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\begin{document}

\extratitle{\centering\"Oklid'in \"O\u gelerinin 13 Kitab\i ndan Birinci Kitap}
\title{\"O\u gelerin 13 Kitab\i ndan\\
Birinci Kitap}
\author{\"Oklid'in Yunanca metni\\
ve\\
\"Ozer \"Ozt\"urk \&\ David Pierce'in \c cevirdi\u gi T\"urk\c cesi}
\date{D\"uzeltilmi\c s 3.\ bask\i\\
\today}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\url{http://mat.msgsu.edu.tr/}
}

\uppertitleback{\centering
Bu \c cal\i\c sma\\
 Creative Commons Attribution-Gayriticari-ShareAlike 3.0\\
Unported Lisans\i\ ile lisansl\i.\\
Lisans\i n bir kopyas\i n\i\ g\"orebilmek i\c cin,\\
\url{http://creativecommons.org/licenses/by-nc-sa/3.0/}\\
adresini ziyaret edin ya da mektup at\i n:\\
Creative Commons,\\
444 Castro Street, Suite 900,\\
Mountain View,
California, 94041, USA.\\
\mbox{}\\
\cc \ccby \"Ozer \"Ozt\"urk \&\ David Pierce \ccnc \ccsa\\
\mbox{}\\
\url{ozer.ozturk@msgsu.edu.tr}\qquad \url{dpierce@msgsu.edu.tr}\\
}

\maketitle[-1]

\chapter*{\"Ons\"oz}

Bu kitapta, \"Oklid'in \emph{\"O\u geler}'inin birinci kitab\i n\i n 
orijinal Yunanca metni ve paralel T\"urk\c ce \c ceviri birlikte sunulmu\c stur.
Kitab\i m\i z,
Mimar Sinan G\"uzel Sanatlar \"Universitesi'nin Matematik B\"ol\"um\"u'nde 
bir birinci s\i n\i f lisans dersi i\c cin haz\i rlanm\i\c st\i r.

Kitab\i n birinci bask\i s\i, 2011 G\"uz d\"oneminde,
ve ikinci bask\i s\i, 2012 G\"uz d\"oneminde kullan\i lm\i\c s
ve fark edilen hatalar d\"uzeltilmi\c stir.  

\.Ilk dersin \"o\u gretmenleri, 
\"Ozer \"Ozt\"urk ve David Pierce oldu; 
sonraki dersin \"o\u gretmenleri, 
Ahmet Bakkalo\u glu, Ayhan G\"unayd\i n, 
\"Ozer \"Ozt\"urk ve David Pierce oldu.

Kitab\i n ilk iki bask\i s\i nda, 
\.Ingilizce \c cevirisi de vard\i.  
Bu \"u\c c\"unc\"u bask\i ya
\.Ingilizce \c ceviriyi almad\i k.

Buradaki Yunanca metin, Heiberg'indir \cite{Euclid-Heiberg}.  
Kitab\i n\i n kopyas\i, internet'te bulunabilir, mesela
Wilbour Hall\footnote{\url{http://www.wilbourhall.org}} ve
European Cultural Heritage Online
(ECHO)\footnote{\url{http://echo.mpiwg-berlin.mpg.de/home}}
sitelerinde.  
Asl\i nda \LaTeX\ elektronik dosyam\i z i\c cin 
Fitzpatrick'in \LaTeX\ kayna\u g\i n\i\ \cite{euclid-fitzpatrick} 
kullanm\i\c st\i k.
Ama Fitzpatrick'in dosyas\i ndaki metni 
Heiberg'in kitab\i ndan nas\i l ald\i\u g\i n\i\ bilmiyoruz,
ve bu metinde 
%a\c sa\u g\i daki 
birka\c c hatalar\i\ fark ettik.\footnote{
\begin{tabular}{|c|ccc|ccc|}
\hline
 &\multicolumn{3}{c|}{Fitzpatrick}&\multicolumn{3}{c|}{Heiberg}\\
\"Onerme&sat\i r&sayfa& & &sayfa&sat\i r\\\hline
5 (\gfs{\greeknumeral 5})&ilk&11&\gfs{tr`oc}&\gfs{pr`oc}&20&2\\
17 (\gfs{\greeknumeral{17}})&2&21&\gfs{p'ant~h|}&\gfs{p'anth|}&44&10\\
17 (\gfs{\greeknumeral{17}})&son&22&\gfs{p'ant~h|}&\gfs{p'anth|}&44&24\\
36 (\gfs{\greeknumeral{36}})& &  &\gfs{d`ia}&\gfs{di`a}&88&4\\
36 (\gfs{\greeknumeral{37}})& &  &\gfs{d`ia}&\gfs{di`a}&88&20\\
38 (\gfs{\greeknumeral{38}})&7&39&\gfs{d`ia}&\gfs{di`a}&90&17\\\hline
  \end{tabular}}
Bu hatalar, Project Perseus sitesinde bulunmamaktad\i
r.\footnote{\url{http://www.perseus.tufts.edu/}}   

Project Perseus sitesinden \c cok faydaland\i k.  G\"uler \c Celgin'in \cite{Celgin} s\"ozl\"u\u g\"u de yararl\i yd\i.  Kulland\i\u g\i m\i z Yunanca font, Greek Font Society (Yunan Font Derne\u gi) taraf\i ndan sa\u glanan ``NeoHellenic'' fontudur. 

%\setcounter{tocdepth}0

\tableofcontents

\addchap{Giri\c s}

Bildi\u gimiz kadar\i\ ile, a\c sa\u g\i\ yukar\i\ bir y\"uzy\i l \"onceye kadar, en az\i ndan D\"unyan\i n Hristiyan ve M\"usl\"uman yerlerinde, her matematikci matemati\u gi \"Oklid'den \"o\u grendi.  Bizce matematik \"o\u grencileri, h\^al\^a \"Oklid'i okumal\i lard\i r.  \emph{\"O\u geler} eseri, d\"unyan\i n ilk matematik dizgesidir.

Her kitap gibi, \"Oklid'in \emph{\"O\u geler}'i m\"ukkemel olmayabilir.  Yap\i s\i nda hatalar varsa, \"o\u grenci onlar\i\ d\"uzelterek \"o\u grensin.  Bug\"unk\"u ``analitik'' geometri ders kitaplar\i, mant\i k a\c c\i s\i ndan d\"uzensiz olabilir, ama \emph{\"O\u geler}'in birinci kitab\i n\i n yard\i m\i yla d\"uzeltilebilir.

%\section*{Sayfa d\" uzeni ve metin}
\section*{Metnimiz}

%\begin{multicols}{2}

\"Oklid'in \emph{\"O\u geler}'inin birinci kitab\i,
burada iki s\"utun halinde sunuluyor: 
sol s\"utunda orijinal Yunanca metin, 
ve sa\u g\i nda bir T\"urk\c ce \c cevirisi yer al\i yor.

\"Oklid'in \emph{\"O\u geler}'i,  
her biri \textbf{\"onermelere} b\"ol\"unm\"u\c s olan 13 kitaptan olu\c sur. 
Baz\i\ kitaplarda  \textbf{tan\i mlar} da vard\i r. 
Birinci kitap ayr\i ca 
\textbf{post\"ulatlar\i} ve \textbf{ortak kavramlar\i} da i\c cerir.  
Bu bask\i da Yunanca metnin her \"onermesinin her c\"umlesi 
\"oyle birimlere b\"ol\"unm\"u\c st\"ur ki
\begin{compactenum}[1)]
\item
(hemen hemen) her birim bir sat\i ra s\i\u gar, 
\item
her birim c\"umle i\c cinde bir rol oynar,
\item
her birimin tam T\"urk\c ce \c cevirisi vard\i r.
\end{compactenum}
Her birimin \c cevirisi, orijinalinin yan\i nda yer al\i r.  
Bazen ortaya \c c\i kan T\"urk\c ce c\"umleler, 
biraz tuhaf gelebilir.
Bu durumda, daha ak\i c\i\ ifadeler bulmak okuyucuya b\i rak\i lm\i\c st\i r.
  
\emph{\"O\u geler}'in her \"onermesinin yan\i nda, 
\c co\u gu noktan\i n (ve baz\i\ \c cizgilerin) harflerle isimlendirildi\u gi, 
bir \c cizgi ve noktalar resmi yer al\i r. 
Bu resim \textbf{harf{}li diagramd\i r}. 
Her \"onermede diagram\i\  kelimelerin \emph{sonuna} yerle\c stiriyoruz. Reviel Netz'e g\"ore orijinal ruloda diagram burada yer al\i rd\i\  ve b\"oylece okuyan \"onermeyi okumak i\c cin ruloyu ne kadar a\c cmas\i\  gerekti\u gini bilirdi ~\cite[p.~35, n.~55]{MR1683176}.  
Bu bask\i da bir \"onerme iki sayfaya s\i\u gmazsa, diagram\i\ tekrarlan\i r.

%\"Oklid'in yazd\i klar\i n\i n \c ce\c sitli s\"uzge\c clerden ge\c cmi\c s haline ula\c sabiliyoruz. 
\"Oklid'in yazd\i klar\i, 
\c ce\c sitli s\"uzge\c clerden ge\c cerek bize ula\c sm\i\c st\i r.
\emph{\"O\u geler}'in M.\"O.\ 300 civar\i nda yaz\i lm\i\c s olmas\i\  gerekir. 
Bizim kulland\i\u g\i m\i z 1883'te yay\i nlanan Heiberg \cite{Euclid-Heiberg} versiyonu, 
10. y\"uzy\i lda yaz\i lm\i\c s ve Vatikan'da bulunmu\c s 
bir elyazmas\i na dayanmaktad\i r.

%\end{multicols}

\section*{Dili ve alfabesi}

\"Oklid'in kulland\i\u g\i\ dil,
Antik Yunancad\i r.
Bu dil,  
\.Ingilizce ve Fars\c ca gibi,
Hint-Avrupa dilleri ailesindendir.  
T\"urk\c ce, bu aileden de\u gildir; 
fakat baz\i\ y\"onlerden T\"urk\c ce, 
Yunancaya, \.Ingilizceden daha yak\i nd\i r. 
\"Orne\u gin T\"urk\c ce ve Yunanca, adlar ve f{}iiller \c ceker.
\.Ingilizce ve T\"urk\c cenin g\"un\"um\"uz bilimsel terminolojisinin k\"okleri 
genellikle Yunancad\i r.

Yunan alfabesinin a\c sa\u g\i daki \pageref{tab:alfabe} numaral\i\ sayfada verilen 24 harf{}ini ezberlemenizi tavsiye ederiz.  
\begin{table}[ht]
  \begin{center}
      \begin{tabular}{c c c c}
b\"uy\"uk&k\"u\c c\"uk&okunu\c s&isim\\\hline
\gfs{A}& \gfs{a} & a & alfa \\ 
\gfs{B}& \gfs{b} & b & beta   \\ 
\gfs{G}& \gfs{g} & g & gamma \\ 
\gfs{D}& \gfs{d} & d & delta \\ 
\gfs{E}& \gfs{e} & e (k\i sa) & epsilon\\ 
%&\gfs{\stigma}& stigma\\
\gfs{Z}& \gfs{z} & z (ds) & zeta   \\ 
\gfs{H}& \gfs{h} & \^e (uzun e) & eta \\ 
 \gfs{J}& \gfs{j} & th & theta \\
%\hline 
 \gfs{I}& \gfs{i} & i & iota (yota)\\ 
\gfs{K}& \gfs{k} & k & kappa\\ 
 \gfs{L}& \gfs{l} & l & lambda\\ 
 \gfs{M}& \gfs{m} & m & m\"u \\
 \gfs{N}& \gfs{n} & n & n\"u \\ 
 \gfs{\Xionealt}& \gfs{x} & ks & ksi \\ 
 \gfs{O}& \gfs{o} & o (k\i sa) & omikron\\ 
 \gfs{P}& \gfs{p} & p & pi\\ 
%&\gfs{\qoppa}& koppa\\
%\hline
 \gfs{R}& \gfs{r} & r & rho (ro)\\ 
 \gfs{S}& \gfs{sv, c} & s & sigma \\
 \gfs{T}& \gfs{t} & t & tau \\ 
 \gfs{U}& \gfs{u} & y, \"u & \"upsilon \\ 
 \gfs{F}& \gfs{f} & f & phi\\ 
 \gfs{Q}& \gfs{q} & h (kh) & khi\\ 
 \gfs{Y}& \gfs{y} & ps & psi\\ 
 \gfs{W}& \gfs{w} & \^o (uzun o)& omega\\
%&\gfs{\sampi}& sampi\\
\hline
%b\"uy\"uk&k\"u\c c\"uk&okunu\c s&isim
  \end{tabular}
  \end{center}
\caption*{Yunan alfabesi}\label{tab:alfabe}
\end{table}
Bu kitapta her \"onermenin sadece bir diagram\i\ vard\i r, 
ve harf{}leri Yunan alfabesinden al\i nm\i\c st\i r.  
Matematik\c ciler, bu harf{}leri her zaman kullan\i rlar.

\section*{\"O\u gelerin ve \"onermelerinin analizi}\label{sect:analiz}

%\begin{multicols}2


\emph{\"O\u geler}'in her \"onermesi 
bir \textbf{problem} veya bir \textbf{teorem} olarak anla\c s\i labilir.  
M.S.\ 320 civar\i nda (yani \"Oklid'den 6 y\"uzy\i l sonra) yazan 
\.Is\-ken\-de\-ri\-ye\-li Pappos bu ayr\i m\i\ 
a\c sa\u g\i daki gibi 
tarif ediyor:%%%%%%%%%%%%%%%%%%%%%%
\footnote{Pappos'tan yap\i lan al\i nt\i, 
onun \emph{Toplama} eserinin \"u\c c\"unc\"u kitab\i n\i n~\cite[s.~30]{Pappus-1} 
giri\c sinden al\i nm\i\c st\i r.  
Al\i nt\i, \cite[pp.~566--567]{MR13:419b} kayna\u g\i nda da bulunabilir.}

\vspace{0.5\baselineskip}
\begin{textpart}\relscale{0.9}
\begin{parprose}
{O<i t`a >en gewmetr'ia| zhto'umena boul'omenoi teqnik'wteron
diakr'inein, 
%>~w kr'atiste Pandros'ion,
}
{%Say\i n Pandrosios:  
Geometri ara\c st\i rmalar\i nda daha usta bir ayr\i\c st\i rma yapmak isteyenler,}
\end{parprose}
\begin{parprose}
{\textbf{pr'oblhma} m`en >axio~usi
kale~in >ef' o<~u \emph{pro\-b'al\-leta'i} ti poi~hsai ka`i kataskeu'asai,}
{bir \c seyin yap\i lmas\i n\i\ veya in\c sa edilmesini \emph{\"oneren} bir [\"onerme]ye \textbf{problem} demeyi uygun g\"or\"uyorlar,}
\end{parprose}
\begin{parprose}{\textbf{je'wrhma} d`e >en <~w| tin~wn <upokeim'enwn t`o <ep'omenon a>uto~is
ka`i p'antws >episumba~inon \emph{jewre~itai,}}
{ve belirli varsay\i mlar\i n e\c sitliklerinin ve zorunlu sonu\c clar\i n\i n incelendi\u gi bir\linebreak {}[\"oner\-me]ye, \textbf{teorem} [demeyi uygun\linebreak g\"or\"uyorlar];}
\end{parprose}
\begin{parprose}
{t~wn palai~wn t~wn m`en
probl'hmata p'anta, t~wn d`e jewr'hmata e>~inai fask'ontwn.}
{ama antiklerin baz\i lar\i\ [\"onermelerin]\linebreak t\"um\"un\"un problem, baz\i lar\i\ da teorem oldu\u gunu s\"oylemi\c stir.}
\end{parprose}
\end{textpart}
Bir problem bir \c sey yapmay\i\ \"onerir; 
bir teorem bir \c sey inceler.
Pappos, \emph{problem} ve \emph{teorem} kelimelerinin etimolojisini an\i\c st\i r\i yor:
\begin{center}
\begin{tabular}{rl|rl}
\gfs{pr'oblhma}&problem&
  \gfs{je'wrhma}&teorem\\
\gfs{proball-}&\"oner-&
\gfs{jewre-}&incele-
\end{tabular}
\end{center}
Bizim \emph{\"onerme} s\"ozc\"u\u g\"um\"uz, Yunanca'da bulunmamaktad\i r, ama etimoloji a\c c\i s\i ndan \gfs{pr'oblhma} ad\i\ gibidir.  Yunan \gfs{jewre-} f{}iili, anlam\i\ ``bak-'' olan \gfs{jea-} f{}iilinden t\"urenmi\c stir.  Bu son f{}iilden \gfs{j'eatron} ``tiyatro'' gelmi\c stir.

\.Ister bir problem, ister bir teorem olsun, 
bir \"onermenin metni alt\i\ par\c caya kadar ayr\i l\i p 
analiz edilebilir. 
M.S.\ be\c sinci y\"uzy\i lda (yani \"Oklid'den 7 y\"uzy\i l sonra)
Proklos bu par\c calar\i\ ve bu analizi anlatm\i\c st\i r:\footnote{Verilen al\i nt\i n\i n Yunancas\i, \cite[s.~203]{1873procli} kayna\u g\i ndan al\i nm\i\c st\i r.  Bu kitab\i n \.Ingilizce \cite{MR1200456} \c cevirisi vard\i r.  Verilen al\i nt\i n\i n \.Ingilizcesi,  \cite[s.~xxiii]{MR1932864} bulunmu\c stur.
Proklos 
  Bizans (\c simdi \.Istanbul) do\u gumludur, ama asl\i nda Likyal\i d\i r, ve ilk e\u gitimini Ksantos'ta alm\i\c st\i r. Felsefe \"o\u grenmek i\c cin \.Iskenderiye'ye ve sonra da Atina'ya  gitmi\c stir \cite[s.~xxxix]{MR1200456}.}

\vspace{0.5\baselineskip}
\begin{textpart}\relscale{0.9}
  \begin{parprose}{p~an d`e
pr'oblhma ka`i p~an je'wrhma t`o >ek tele'iwn t~wn <eauto~u
mer~wn sumpeplhrwm'en\-on bo'uletai p'anta ta~uta >'eqein
{}>en <eaut~w|;}
{B\"ut\"un par\c calar\i yla donat\i lm\i\c s her\linebreak problem ve her teorem a\c sa\u g\i daki t\"um par\c calar\i\ i\c cermek ister:}
\end{parprose}
\begin{parprose}{\begin{inparaenum}[{[}i{]}]
\item
\emph{pr'otasin,}
\item
\emph{>'ekjesin,}\\
\item
\emph{diorism'on,}
\item
\emph{kataskeu\-'hn,}\\
\item
\emph{>ap'odeixin,}
\item
\emph{sump'erasma.}
\end{inparaenum}}
{\begin{inparaenum}[(1)]
\item
\emph{bildirme,}
\item
\emph{a\c c\i klama,}\\
\item
\emph{belirtme,}
\item
\emph{d\"uzenleme,}\\
\item
\emph{g\"osterme,} ve 
\item
\emph{bitirme.}
\end{inparaenum}}
\end{parprose}
\begin{parprose}{to'utwn d`e}{Bunlardan da:}
\end{parprose}
\begin{parprose}{ <h m`en \textbf{pr'otasis}
l'egei, t'inos dedom'enou t'i t`o zhto'umen'on >estin.}
{1. \textbf{Bildirme,}
hangi verilenden hangi [sonucun] arand\i\u g\i n\i\ s\"oyler.}
\end{parprose}
\begin{parprose}
{<h g`ar
tele'ia pr'otasis >ex >amfot'erwn >est'in.}
{\Gar\ tam bir bildirme, 
bu iki par\c can\i n ikisini de i\c cerir.}
\end{parprose}
\begin{parprose}{<h d> \textbf{>'ekjesis}
a>ut`o kaj> a<ut`o t`o dodem'enon >apodialabo~usa pro%
eutrep'i\-zei t~h| zht'hsei.}
{2. \textbf{A\c c\i klama,} verileni ayr\i ca ele alarak bunu ara\c st\i rmada kullanmak \"uzere haz\i rlar.}
\end{parprose}
\begin{parprose}{<o d`e \textbf{diorism`os} qwr`is t`o zhto'u%
menon, <'oti pot'e >estin, diasafe~i.}
{3. \textbf{Belirtme,} aranan\i n ayr\i ca ne oldu\u gunu net bir \c sekilde g\"osterir.}
\end{parprose}
\begin{parprose}{<h d`e \textbf{kataskeu`h} t`a
{}>elle'iponta t~w|\linebreak dedom'enw| pr`os t`hn to~u zhtoum'enou j'h%
ran prost'ijhsin.}
{4. \textbf{D\"uzenleme,} aranan\i\ avlamak i\c cin 
verilendeki eksikleri yerle\c smi\c stir.%
%elde edilecek sonuca ula\c smak i\c cin verilende neyin eksik oldu\u gunu s\"oyler.
}
\end{parprose}
\begin{parprose}{<h d`e \textbf{>ap'odeixis} >episthmonik~ws >ap`o
t~wn <omologhj'entwn sun'agei t`o proke'imenon.}
{5. \textbf{G\"osterme,} [elimizde] bulunanlar\i\ bilimsel olarak kabul edilen [ilkeler]e g\"ore birle\c stirir.%
%\"onerilen \c c\i kar\i m\i\ kabul edilen \"onermelerden bilimsel ak\i l y\"ur\"utmeyle olu\c sturur.
}
\end{parprose}
\begin{parprose}{t`o d`e
\textbf{sump'erasma} p'alin >ep`i t`hn pr'otasin >anastr'efei be%
baio~un t`o dedeigm'enon.}
{6. \textbf{Bitirme,} g\"osterilmi\c s olan\i\ onaylayarak
bildirmeye geri d\"oner.}
\end{parprose}
\begin{parprose}{ka`i t`a m`en s'umpanta m'erh
t~wn te problhm'atwn ka`i t~wn jewrhm'atwn >est`i to%
s\-a~uta;}
{Bunlar, problemlerin ve teoremlerin\linebreak b\"ut\"un par\c calar\i d\i r.}
\end{parprose}
\begin{parprose}
{t`a d`e >anagkai'otata ka`i >en p~asin <up'arqon\-ta
pr'otasis ka`i >ap'odeixis ka`i sum\-p'erasma.}
{En zorunlu olan ve her [\"onerme]de bulunan [par\c calar],
bildirme, g\"osterme,\linebreak ve bitirmedir.}
\end{parprose}
\end{textpart}
    
Biz de Proklos'un analizini a\c sa\u g\i daki anlam\i yla kullanaca\u g\i z:
\begin{compactenum}
\item
\emph{Bildirme,} bir \"onermenin, harf{}li diagrama g\"onderme yapmayan, genel beyan\i d\i r.  Bu beyan, bir do\u gru veya \"u\c cgen gibi bir nesne hakk\i ndad\i r.   
\item
\emph{A\c c\i klama,} bu nesneyi harf{}ler arac\i l\i\u g\i yla diagramda i\c saret eder.  
Bu nesnenin varl\i\u g\i\ \"u\c c\"unc\"u tekil emir kipinde bir f{}iil ile olu\c sturulur. 
\item
\emph{Belirtme,} 
\begin{compactenum}
\item    
bir \emph{problemde}, 
nesne ile ilgili ne yap\i laca\u g\i n\i\ s\"oyler 
ve \gfs{de~i d`h} kelimeleriyle ba\c slar 
(burada \gfs{de~i}, ``\dei'', \gfs{d'h} ise ``\dee'' anlam\i ndad\i r);
\item
bir \emph{teoremde,} nesneyle ilgili neyin ispatlanaca\u g\i n\i\ s\"oyler ve ``\legohoti'' anlam\i na gelen \gfs{l'egw <'oti} kelimeleriyle ba\c slar. 
Ayn\i\ ifade, bir problemde de belirtmeye ek olarak, 
g\"ostermenin ba\c s\i nda ve
d\"uzenlemenin sonunda g\"or\"ulebilir. 
\end{compactenum}
\item
 \emph{D\"uzenleme} varsa, ikinci kelimesi 
\gfs{g'ar} olur.  Bu kelime, onaylay\i c\i\ bir zarf ve sebep belirten bir ba\u gla\c ct\i r. Bunu ``\gar'' olarak \c cevirdik ve c\"umlenin birinci kelimesi yapt\i k.
\item
\emph{G\"osterme,} genellikle \gfs{>epe'i} (``\c c\"unk\"u,
oldu\u gundan'') ilgeciyle ba\c slar.   
\item
\emph{Bitirme,} bildirmeyi tekrarlar ve genellikle \gfs{>'ara} (``\ara'') ilgecini i\c cerir.  Tekrarlanan bildirmeden sonra bitirme a\c sa\u g\i daki iki kal\i ptan biriyle sonlan\i r:
\begin{compactenum}
\item
\gfs{<'oper >'edei poi~hsai} ``\ozqef'' (problemlerde; Latincesi \emph{quod erat faciendum} veya QEF); 
\item
\gfs{<'oper >'edei de~ixai} ``\ozqed'' (teoremlerde; Latincesi \emph{quod erat demonstrandum} veya QED).
\end{compactenum}
\end{compactenum}

%\end{multicols}



 

%\end{multicols}



%\input{euclid-introduction}
%\input{euclid-introduction-tr3}

\addchap{\gfs{<'Oroi} // Hudutlar}

\begin{textpart}

\newparsen{
Shme~i'on >estin,\\
o<~u m'eroc o>uj'en.
}
{
{}[1] Bir \textbf{nokta,}\\
hi\c cbir par\c cas\i\ olmayand\i r.
}

\newparsen{
Gramm`h d`e\\
m~hkoc >aplat'ec.
}
{
{}[2] Ve bir \textbf{\c cizgi,}\\
geni\c sliksiz uzunluktur.
}

\newparsen{
Gramm~hc d`e\\
p'erata shme~ia.
}
{
{}[3] Ve bir \c cizginin\\
s\i n\i rlar\i, noktad\i r.
}

\newparsen{
E>uje~ia gramm'h >estin,\\
<'htic >ex >'isou\\
to~ic >ef> <eaut~hc shme'ioic\\
ke~itai%
\eix{\gr{ke~imai}!\gr{ke~itai} 3rd sg pres ind mp}.
}
{
{}[4] Bir \textbf{do\u gru} \c cizgi,\\
e\c sit olarak\\
\"uzerindeki noktalara g\"ore\\
oturand\i r.\footnotemark
}\footnotetext{Lucio Russo'ya \cite[s.~322--4]{MR2038833} g\"ore bu tan\i m ve buradaki ba\c ska tan\i mlar, \emph{Heron'un Tan\i mlar\i} (\emph{Heronis Definitiones}) adl\i\ kitab\i ndan \"Oklid'in \emph{\"O\u geler}'ine eklenmi\c stir.  \emph{Heron'un Tan\i mlar\i}'nda \gfs{E>uje~ia m`en o>~un gramm'h >estin
<'htic >ex >'isou
to~ic >ep> a>ut~hc shme'ioic
ke~itai >orj`h o>~usa ka`i o<~ion >ep> >'akron tetam'enh >ep`i t`a p'erata} ``Bir do\u gru \c cizgi,
e\c sit olarak
\"uzerindeki noktalara g\"ore
d\"uz ve u\c clar\i ndan en fazla gerilmi\c s
oturand\i r'' (\emph{A straight line is a line that equally with respect to [all] points on itself lies straight and maximally taught between its extremities}) metni bulunmu\c stur.}
%\stepcounter{footnote}

\newparsen{
>Epif'aneia d'e >estin,\\
<`o m~hkoc \kai%
\eix{\gr{ka`i}} pl'atoc m'onon\\
{}>'eqei%
\eix{\gr{>'eqw}}.
}
{
{}[5] Ve bir \textbf{y\" uzey,}\\
sadece uzunlu\u gu ve geni\c sli\u gi\\
oland\i r.
}

\newparsen{
>Epifane'iac d`e\\
p'erata  gramma'i.
}
{
{}[6] Ve bir y\" uzeyin\\
s\i n\i rlar\i, \c cizgidir.
}

\newparsen{
>Ep'ipedoc >epif'anei'a >estin,\\
<'htic >ex >'isou\\
ta~ic >ef> <eaut~hc e>uje'iaic\\
ke~itai%
\eix{\gr{ke~imai}!\gr{ke~itai} 3rd sg pres ind mp}.
}
{
{}[7] Bir \textbf{d\" uzlem} y\"uzeyi,\\
e\c sit olarak\\
\"uzerindeki do\u grulara g\"ore\\
oturand\i r.
}

\newparsen{
>Ep'ipedoc d`e gwn'ia >est`in\\
<h >en >epip'edw|\\
d'uo gramm~wn <aptom'enwn%
\eix{\gr{<'aptw}!\gr{<apt'omenoc} part pres mp} 
>all'hlwn\\
\kai%
\eix{\gr{ka`i}} m`h >ep> e>uje'iac keim'enwn%
\eix{\gr{ke~imai}!\gr{ke'imenoc} part pres mp}\\
pr`oc >all'hlac t~wn gramm~wn\\
kl'isic.
}
{
{}[8] Ve bir \textbf{d\" uzlem a\c c\i s\i,}\\
bir d\"uzlemde\\
iki \c cizgi birbirine dokununca\\
ve bir do\u gru \"uzerinde oturmay\i nca\\
\c cizgilerin birbirine g\"ore\\
e\u gimidir.
}


\newparsen{
<'Otan d`e a<i peri'eqousai%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act} t`hn gwn'ian\\
gramma`i\\
e>uje~iai >~wsin,\\
e>uj'ugrammoc kale~itai%
\eix{\gr{kal'ew}} 
<h gwn'ia.
}
{
{}[9] Ve ne zaman a\c c\i y\i\ i\c ceren\\
\c cizgiler\\
do\u gru olursa\\
a\c c\i ya \textbf{d\"uzkenar} denir.
}

\newparsen{
<'Otan d`e e>uje~ia\\
{}>ep> e>uje~ian staje~isa%
\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
t`ac >efex~hc gwn'iac\\
{}>'isac >all'hlaic poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
{}>orj`h <ekat'era t~wn >'iswn gwni~wn >esti,\\
\kai%
\eix{\gr{ka`i}} <h >efesthku~ia%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>efesthk'wc} part perf act} 
e>uje~ia\\
k'ajetoc kale~itai%
\eix{\gr{kal'ew}},\\
{}>ef> <`hn >ef'esthken%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>ef'esthke} 3rd sg perf ind act}.
}
{
{}[10] Ve ne zaman bir do\u gru,\\
bir do\u grunun \"uzerine dikilmi\c s,\\
biti\c sik a\c c\i lar\i\\
birbirine e\c sit yaparsa,\\
e\c sit a\c c\i lar\i n her biri, \textbf{diktir,}\\
ve dikilmi\c s do\u gruya\\
\textbf{dikey} denir\\
\"uzerine dikildigi [do\u gru]ya.\footnotemark
}
\footnotetext{Bu tan\i m, 11.\ ve 12.\ \"onermelerde al\i nt\i lan\i r.}

\newparsen{
>Amble~ia gwn'ia >est`in\\
<h me'izwn >orj~hc.
}
{
{}[11] Bir \textbf{geni\c s a\c c\i,}\\
dik [a\c c\i]dan b\"uy\"uk oland\i r.\footnotemark 
}\footnotetext{Atat\"urk'\"un \emph{Geometri} kitab\i na \cite[\P 37, s.~15]{Ataturk} g\"ore \"oyle bir a\c c\i, \textbf{oput a\c c\i d\i r.}}

\newparsen{
>Oxe~ia d`e\\
<h >el'asswn >orj~hc.
}
{
{}[12] Ve bir \textbf{dar a\c c\i,}\\
dik [a\c c\i]dan k\"u\c c\"uk oland\i r. 
}

\newparsen{
<'Oroc >est'in,\\
<'o tin'oc >esti p'erac.
}
{
{}[13] Bir \textbf{hudut,}\\
herhangi bir \c seyin s\i n\i r\i\ oland\i r.
}

\newparsen{
Sq~hm'a >esti\\
t`o <up'o tinoc >'h tinwn <'orwn\\
perieq'omenon%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp}.
}
{
{}[14] Bir \textbf{f{}ig\"ur,}\\
bir hudut veya hudutlar taraf\i ndan\\
i\c cerilendir. 
}

\newparsen{
K'ukloc >est`i\\
sq~hma >ep'ipedon\\
<up`o mi~ac gramm~hc perieq'omenon%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp}\\
{}[<`h kale~itai%
\eix{\gr{kal'ew}} perif'ereia],\\
pr`oc <`hn\\
{}>af> <en`oc shme'iou\\
t~wn >ent`oc to~u sq'hmatoc keim'enwn%
\eix{\gr{ke~imai}!\gr{ke'imenoc} part pres mp}\lli\\
p~asai a<i prosp'iptousai%
\eix{\gr{p'iptw}!\gr{prosp'iptw}!\gr{prosp'iptwn} part pres act} 
e>uje~iai\\
{}[pr`oc t`hn to~u k'uklou perif'ereian]\\
{}>'isai >all'hlaic e>is'in.
}
{
{}[15] Bir \textbf{daire,}\\
d\"uzlemdeki bir f{}ig\"urd\"ur\\
bir \c cizgice i\c cerilen\\
{}[bu \c cizgiye \textbf{\c cevre} denir]\\
\"oyle ki [bu \c cizginin \"uzerine]\\
bir noktas\i ndan\\
(f{}ig\"ur\"un i\c cerisinde oturan
 noktalar\i n)\\
t\"um d\"u\c sen do\u grular,\\
{}[\c cevrenin \"uzerine]\\
birbirine e\c sittir.
}

\newparsen{
K'entron d`e to~u k'uklou\\
t`o shme~ion kale~itai%
\eix{\gr{kal'ew}}.
}
{
{}[16] Ve dairenin \textbf{merkezi}\\
denir o noktaya.
}

\newparsen{
Di'ametroc d`e to~u k'uklou >est`in\\
e>uje~i'a tic\\
di`a to~u k'entrou >hgm'enh%
\eix{\gr{>'agw}!\gr{>hgm'enoc} part perf mp}\\
\kai%
\eix{\gr{ka`i}} peratoum'enh%
\eix{\gr{perat'ow}!\gr{peratoum'enoc} part pres mp}\\
{}>ef> <ek'atera t`a m'erh\\
<up`o t~hc to~u k'uklou perifere'iac,\\
<'htic \kai%
\eix{\gr{ka`i}}\\
d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}
t`on k'uklon.
}
{
{}[17] Ve bir dairenin bir \textbf{\c cap\i,}\\
herhangi bir do\u grudur\\
dairenin merkezinden ilerletilmi\c s\\
ve s\i n\i rland\i r\i lan\\
 her iki tarafta\\
 dairenin \c cevresi taraf\i ndan;\\ 
ve [b\"oyle bir do\u gru,]\\
daireyi ikiye b\"oler.\\
}

\newparsen{
<Hmik'uklion d'e >esti\\
t`o perieq'omenon%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp} 
sq~hma\\
<up'o te%
\eix{\gr{te}} t~hc diam'etrou\\
\kai%
\eix{\gr{ka`i}} t~hc >apolambanom'enhc%
\eix{\gr{lamb'anw}!\gr{>apolamb'anw}!\gr{>apolamban'omenoc} part pres mp} 
<up> a>ut~hc\\
perifere'iac.\\
k'entron d`e to~u <hmikukl'iou t`o a>ut'o,\\
<`o \kai%
\eix{\gr{ka`i}} to~u k'uklou >est'in.
}
{
{}[18] Bir \textbf{yar\i daire,}\\
i\c cerilen f{}ig\"urd\"ur\\
hem bir \c cap\\
hem onun ay\i rd\i\u g\i\\
\c cevre taraf\i ndan.\\
Ve yar\i dairenin merkezi ayn\i d\i r\\
daireninkiyle. 
}

\newparsen{
Sq'hmata e>uj'ugramm'a >esti\\
t`a <up`o e>ujei~wn perieq'omena%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp},\\
tr'ipleura m`en%
\eix{\gr{m'en}} t`a <up`o tri~wn,\\
tetr'apleura d`e t`a <up`o tess'arwn,\\
pol'upleura d`e\\
t`a <up`o plei'onwn >`h tess'arwn\\
e>ujei~wn perieq'omena%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp}.
}
{
{}[19] \textbf{D\"uzkenar f{}ig\"urler,}\\
do\u grularca i\c cerilendir:\\
\textbf{\"u\c ckenar} f{}ig\"urler \"u\c c,\\
\textbf{d\"ortkenar} f{}ig\"urler de d\"ort,\\
\textbf{\c cokkenar} f{}ig\"urler de\\
d\"ortten daha fazla\\
do\u gruca i\c cerilendir. 
}

\newparsen{
T~wn d`e triple'urwn sqhm'atwn\\
{}>is'opleuron m`en%
\eix{\gr{m'en}} tr'igwn'on >esti\\
t`o t`ac tre~ic >'isac >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} pleur'ac,\\
{}>isoskel`ec d`e\\
t`o t`ac d'uo m'onac >'isac >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} pleur'ac,\\
skalhn`on d`e\\
t`o t`ac tre~ic >an'isouc >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} pleur'ac.
}
{
{}[20] Ve \"u\c ckenar f{}ig\"urlerden\\
\textbf{e\c skenar} \"u\c cgen,\\
\"u\c c e\c sit kenar\i\ olan;\\
\textbf{ikizkenar} da,\\
sadece iki e\c sit kenar\i\ olan;\\
\textbf{\c ce\c sitkenar} da,\\
\"u\c c e\c sit olmayan kenar\i\ oland\i r.
}

\parsen{
{}[21] Yet of trilateral figures,\\
a right-angled triangle is\\
that having a right angle,\\
obtuse-angled, having an obtuse angle,\\
acute-angled, having three acute angles.
}
{
>'Eti%
\eix{\gr{>eti}} 
d`e t~wn triple'urwn sqhm'atwn\\
{}>orjog'wnion m`en%
\eix{\gr{m'en}} tr'igwn'on >esti\\
t`o >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} >orj`hn gwn'ian,\\
{}>amblug'wnion d`e\\
t`o >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} >amble~ian gwn'ian,\\
{}>oxug'wnion d`e\\
t`o t`ac tre~ic >oxe'iac >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} gwn'iac.
}
{
{}[21] Ve ayr\i ca, \"u\c ckenar f{}ig\"urlerden,\\
\textbf{dik [a\c c\i l\i]} \"u\c cgen,\\
bir dik a\c c\i s\i\ olan;\\
\textbf{geni\c s a\c c\i l\i} da,\\
bir geni\c s a\c c\i s\i\ olan;\\
\textbf{dar a\c c\i l\i} da,\\
\"u\c c dar a\c c\i s\i\ oland\i r.
}

\newparsen{
T`wn d`e tetraple'urwn sqhm'atwn\\
tetr'agwnon m'en >estin,\\
<`o >is'opleur'on t'e >esti\\
\kai%
\eix{\gr{ka`i}} >orjog'wnion,\\
<eter'omhkec d'e,\\
<`o >orjog'wnion m'en,\\
o>uk >is'opleuron d'e,\\
<r'omboc d'e,\\
<`o >is'opleuron m'en,\\
o>uk >orjog'wnion d'e,\\
<romboeid`ec d`e\\
t`o t`ac >apenant'ion pleur'ac\\
te \kai%
\eix{\gr{ka`i}} gwn'iac >'isac >all'hlaic >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act},\\
<`o o>'ute >is'opleur'on >estin\\
o>'ute >orjog'wnion;\\
t`a d`e par`a ta~uta\\
tetr'apleura\\
trap'ezia kale'isjw%
\eix{\gr{kal'ew}!\gr{kale'isjw} 3rd sg pres imperat mp}.
}
{
{}[22] Ve d\"ortkenar f{}ig\"urlerden\\
\textbf{kare,}\\
hem e\c skenar olan\\
hem dik;\\
\textbf{dikd\"ortgen} de\\
dik olan\\
ama e\c skenar olmayan;\\
\textbf{romb}\footnotemark\ da,\\
e\c skenar olan\\
ama dik olmayan;\\
\textbf{romboid} de\\
hem kar\c s\i l\i kl\i\ kenar\\
hem a\c c\i lar\i\ e\c sit olan\\
ama ne e\c skenar\\
ne dik oland\i r.\\
Ve bunlar\i n d\i\c s\i nda \kalan\\
d\"ortkenarlara\\
\textbf{trapezion}\footnotemark\ denilsin.
}
\addtocounter{footnote}{-1}
\footnotetext{Yani \emph{e\c skenar d\"ortgen.}}
\stepcounter{footnote}
\footnotetext{\emph{Romb} ve \emph{romboid} terimleri, \"onermelerde kullan\i lmaz.  \emph{Trapezion} terimi, 35.\ \"onermede, yamuk i\c cin kullan\i l\i r.}

\newparsen{
Par'allhlo'i e>isin e>uje~iai,\\
a<'itinec {}>en t~w| a>ut~w| >epip'edw| o>~usai\\
\kai%
\eix{\gr{ka`i}} >ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp} 
e>ic >'apeiron\\ 
{}>ef> <ek'atera t`a m'erh\\
{}>ep`i mhd'etera\\
sump'iptousin%
\eix{\gr{p'iptw}!\gr{sump'iptw}} >all'hlaic.
}
{
{}[23] \textbf{Paraleldir} do\u grular,\\
ayn\i\ d\"uzlemde bulunan\\
ve sonsuza uzat\i l\i nca\\
her iki tarafta,\\
hi\c cbir tarafta\\
\c carp\i\c smayan. 
}


\end{textpart}

\ifthenelse{\isodd{\value{page}}}{}{\newpage\mbox{}}  % So that
                                % postulates and axioms will be on
                                % facing pages  

\addchap{\gfs{A>it'hmata} // Postulatlar}

\begin{textpart}

\newparsen{
>Hit'hsjw%
\eix{\gr{>ait'ew}!\gr{>eit'hsjw} 3rd sg perf imperat mp}\\
{}>ap`o pant`oc shme'iou\\
{}>ep`i p~an shme~ion\\
e>uje~ian gramm`hn\\
{}>agage~in%
\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
{}[Postulat olarak] rica edilmi\c s olsun:\\
{}[1] herhangi bir noktadan\\
herhangi bir noktaya\\
bir do\u gru \c cizgi\\
ilerletmek.
}

\parsen{
Also, a bounded \strgt\\
continuously\\
in a straight\\
to extend.
}
{
\kai%
\eix{\gr{ka`i}} peperasm'enhn%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp} e>uje~ian\\
kat`a t`o suneq`ec%
\eix{\gr{>'eqw}!\gr{sun'eqw}!\gr{suneq'hc} adj}\\
{}>ep> e>uje'iac\\
{}>ekbale~in%
\eix{\gr{>ekb'allw}!\gr{>ekbale~in} aor inf act}.
}
{
{}[2] Ve s\i n\i rlanm\i\c s bir do\u gruyu\\
 kesiksiz \c sekilde\\
 bir do\u gruda\\
 uzatmak.
}

\parsen{
Also, to any center\\
and distance\\
a circle\\
to draw.
}
{
\kai%
\eix{\gr{ka`i}} pant`i k'entrw|\\
\kai%
\eix{\gr{ka`i}} diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}}\\
k'uklon\\
gr'afesjai%
\eix{\gr{gr'afw}!\gr{gr'afesjai} pres inf mp}.
}
{{}[3] Ve her merkez\\
 ve uzunlu\u ga\\
 bir daire\\
 \c cizmek.
}

\parsen{
Also, all right angles\\
equal to one another\\
to be.
}
{
\kai%
\eix{\gr{ka`i}} p'asac t`ac >orj`ac gwn'iac\\
{}>'isac >all'hlaic
e>~inai.
}
{{}[4] Ve b\"ut\"un dik a\c c\i lar\i n\\
birbirine e\c sit
 oldu\u gu.
}

\parsen{
Also, if in two straight lines\\
falling\\
the interior angles to the same parts\\
less than two \rgt s make,\\
the two \strgt s, extended\\
to infinity,\\
fall together,\\
to which parts are\\
the less than two \rgt s.
}
{
\kai%
\eix{\gr{ka`i}} >e`an e>ic d'uo e>uje'iac\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
gwn'iac\\
d'uo >orj~wn >el'assonac poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
{}>ekballom'enac%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}\\
t`ac d'uo e>uje'iac\\
{}>ep> >'apeiron\\
sump'iptein,\\
{}>ef> <`a m'erh e>is`in
a<i t~wn d'uo >orj~wn >el'assonec.
}
{{}[5] Ve e\u ger iki do\u grunun \"uzerine\\
 d\"u\c sen bir do\u gru\\
 ayn\i\ tarafta olu\c sturdu\u gu i\c c\\
  a\c c\i lar\i\\
 iki dik a\c c\i dan k\"u\c c\"uk yaparsa,\\
 uzat\i ld\i klar\i nda\\
 bu iki do\u grunun\\
 s\i n\i rs\i zca\\
 \c carp\i\c saca\u g\i,\\
 a\c c\i lar\i n
 iki dik a\c c\i dan k\"u\c c\"uk oldu\u gu tarafta.
}

\end{textpart}

\addchap[\gfs{Koina`i >'ennoiai} // Ortak kavramlar]{\gfs{Koina`i >'ennoiai} // Ortak kavramlar\footnotemark}
\footnotetext{\emph{Ortak kavram} ad\i n\i n yerine \emph{aksiyom} kullan\i labilir.}

\begin{textpart}

\parsen{
Equals to the same\\
also to one another are equal.
}
{
T`a t~w| a>ut~w| >'isa\\
\kai%
\eix{\gr{ka`i}} >all'hloic >est`in >'isa.
}
{{}[1] Ayn\i\ \c seye e\c sitler\\
  birbirine de e\c sittir.\footnotemark
}\footnotetext{Bu c\"umle, 1., 2., ve 13.\ \"onermelerde al\i nt\i lan\i r.}
%\stepcounter{footnote}

\parsen{
Also, if to equals\\
equals be added,\\
the wholes are equal.
}
{
\kai%
\eix{\gr{ka`i}} >e`an >'isoic\\
{}>'isa prostej~h|,\\
t`a <'ola >est`in >'isa.
}
{{}[2] Ve e\u ger e\c sitlere\\
 e\c sitler eklenirse,\\ 
%elde edilenler 
b\"ut\"unler
e\c sittir.
}

\parsen{
Also, if from equals\\
equals be taken away,\\
the remainders are equal.
}
{
\kai%
\eix{\gr{ka`i}} >e`an >ap`o >'iswn\\
{}>'isa >afairej~h|%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afairej~h} 3rd sg aor subj pass}%
\glossary{\gr{>afairej~h|}},\\
t`a kataleip'omen'a >estin >'isa.
}
{{}[3] Ve e\u ger e\c sitlerden\\ 
e\c sitler ayr\i l\i rsa,\\ 
kalanlar e\c sittir. 
}

\parsen{
Also things applying to one another\\
are equal to one another.
}
{
\kai%
\eix{\gr{ka`i}} t`a >efarm'ozonta%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'ozwn} part pres act}
>ep> >all'hla\\
{}>'isa >all'hloic >est'in.
}
{{}[4] Ve birbirine uygula\c san\footnotemark\ \c seyler\\ 
birbirine e\c sittir.
}\footnotetext{Veya \emph{birbiriyle \c cak\i\c san.}}

\parsen{
Also, the whole\\
than the part is greater.
}
{
\kai%
\eix{\gr{ka`i}} t`o <'olon\\
to~u m'erouc me~iz'on [>estin].
}
{{}[5] Ve b\"ut\"un,\\ par\c cadan b\"uy\"ukt\"ur.
}

\end{textpart}

\addchap{\"Onermeler}

%\addtocontents{toc}{\protect\begin{multicols}{2}}

\begin{proposition}{1}%Proposition I.1
 
\newparsen{
{}>Ep`i
t~hc doje'ishc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass}\\
e>uje'iac peperasm'enhc%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
tr'igwnon >is'opleuron\\
sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}.
}
{
Verilmi\c s\\
s\i n\i rlanm\i\c s\ do\u grunun \"uzerinde\\
e\c skenar \"u\c cgen\\
in\c sa etmek.
}

\newparsen{
>'Estw\\
<h doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia peperasm'enh%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
<h AB.
}
{
Olsun\\
verilmi\c s\ s\i n\i rlanm\i\c s\ do\u gru\\ 
\gk{AB}.
}

\newparsen{
De~i%
\eix{\gr{de~i}}
d`h%
\eix{\gr{d'h}}\\
{}>ep`i t~hc AB e>uje'iac\\
tr'igwnon >is'opleuron\\
sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}.
}
{
\Deidee\\
\gk{AB} do\u grusuna\\
e\c skenar \"u\c cgen\\
in\c sa etmek.
}

\newparsen{
K'entrw| m`en%
\eix{\gr{m'en}} t~w| A\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}}
 d`e t~w| AB\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp}\\
<o BGD,\\
\kai%
\eix{\gr{ka`i}} p'alin%
\eix{\gr{p'alin}}\\
k'entrw| m`en%
\eix{\gr{m'en}} t~w| B\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| BA\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp}\\
<o AGE,\\
\kai%
\eix{\gr{ka`i}} >ap`o to~u G shme'iou,
kaj> <`o t'emnousin%
\eix{\gr{t'emnw}!\gr{t'emnousin} 3rd pl pres ind act}
>all'hlouc o<i k'ukloi,\\ 
{}>ep'i t`a A, B shme~ia\\
{}>epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp}\\ 
e>uje~iai a<i GA, GB.
}
{
\gk{A} merkezine,\\
\gk{AB} uzakl\i \u g\i nda olan\\
daire \c cizilmi\c s\ olsun,\\
\gk{BGD},\\
ve \palin\\
\gk{B} merkezine,\\
\gk{BA} uzakl\i \u g\i nda olan\\
daire \c cizilmi\c s\ olsun,\\
\gk{AGE},\\
ve dairelerin kesi\c sti\u gi
\gk{G} noktas\i ndan\lli\\
\gk{A}, \gk{B} noktalar\i na\\
birle\c stirilmi\c s\ olsun\\
\gk{GA}, \gk{GB} do\u grular\i.
}

\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} t`o A shme~ion
k'entron >est`i to~u GDB k'uklou,\\
{}>'ish >est`in <h AG t~h| AB; \\
p'alin\eix{\gr{p'alin}},\\
{}>epe`i%
\eix{\gr{>epe'i}} t`o B shme~ion
k'entron >est`i to~u GAE k'uklou,\\
{}>'ish >est`in <h BG t~h| BA. \\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}} <h GA t~h|  AB >'ish;\lli\\
<ekat'era >'ara\eix{\gr{>'ara}} t~wn GA, GB t~h| AB
{}>estin >'ish. \\
t`a d`e t~w| a>ut~w| >'isa\\
\kai%
\eix{\gr{ka`i}} >all'hloic >est`in >'isa;\\
\kai%
\eix{\gr{ka`i}} <h GA >'ara\eix{\gr{>'ara}} t~h| GB >estin >'ish;\\
a<i tre~ic >'ara\eix{\gr{>'ara}} a<i GA, AB, BG\\
{}>'isai >all'hlaic e>is'in.
}
{
Ve \gk{A} noktas\i\ \gk{GDB} dairesinin merkezi oldu\u gundan,\\
\gk{AG}, \gk{AB}'ya e\c sittir.\\
\Palin\\
\gk{B} noktas\i\ \gk{GAE} dairesinin merkezi oldu\u gundan,\\
\gk{BG}, \gk{BA}'ya e\c sittir.\\
Ve \gk{GA}'n\i n \gk{AB}'ya e\c sit oldu\u gu g\"osterilmi\c sti.\\
\Ara\ \gk{GA} ile \gk{GB}'n\i n her biri \gk{AB}'ya e\c sittir.\\
Ama ayn\i\ \c seye e\c sitler\\
birbirine de e\c sittir.\\
\Ara\ \gk{GA} da, \gk{GB}'ya e\c sittir.\\
\Ara\ o \"u\c c do\u gru, \gk{GA}, \gk{AB}, \gk{BG},\\
birbirine e\c sittir.
}

\newparsen{
>Is'opleuron >'ara\eix{\gr{>'ara}}\\
{}>est`i t`o ABG tr'igwnon. \\
\kai%
\eix{\gr{ka`i}} sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp}\\
{}>ep`i t~hc doje'ishc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'iac peperasm'enhc%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}
t~hc AB.\\
<'oper >'edei poi~hsai.
}
{
\Ara\ e\c skenard\i r\\
\gk{ABG} \"u\c cgeni.\\
Ve in\c sa edilmi\c stir\\
 verilmi\c s\ s\i n\i rlanm\i\c s
 \gk{AB} do\u grusuna;\lli\\
\ozqef.
}
\vfill
\begin{center}

\psset{unit=2cm}
\begin{pspicture}(3,2)
\pscircle(1,1)1
\pscircle(2,1)1
\uput[l](1,1){\gk A}
\uput[r](2,1){\gk B}
\uput[u](1.5,1.866){\gk G}
\uput[l](0,1){\gk D}
\uput[r](3,1){\gk E}
\pspolygon(1,1)(2,1)(1.5,1.866)
\end{pspicture}
\end{center}
\vfill
\end{proposition}


\begin{proposition}{2}%Proposition I.2

\newparsen{
Pr`oc t~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iw|\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| >'ishn\\
e>uje~ian j'esjai\eix{\gr{t'ijhmi}!\gr{j'esjai} aor inf mp}.
}
{
Verilmi\c s\ noktaya\\
verilmi\c s\ do\u gruya e\c sit olan\\
do\u gru yerle\c stirmek.
}

\parsen{
Let be\\
the given point \gk A,\\
and the given {\strgt}, \gk{BG}.
}
{
>'Estw\\
t`o m`en%
\eix{\gr{m'en}} doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme~ion t`o A,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h BG;
}
{
Olsun\\
verilmi\c s\ nokta \gk{A},\\
verilmi\c s\ do\u gru da \gk{BG}.
}

\parsen{
It is necessary then\\
at the point \gk A\\
equal to the given {\strgt} \gk{BG}\\
for a {\strgt} to be placed.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
pr`oc t~w| A shme'iw|\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| BG >'ishn\\
e>uje~ian j'esjai\eix{\gr{t'ijhmi}!\gr{j'esjai} aor inf mp}.
}
{
\Deidee\\
\gk{A} noktas\i na,\\
verilmi\c s \gk{BG} do\u grusuna e\c sit olan\\
bir do\u gru yerle\c stirmek.
}

\parsen{
For, suppose there has been joined\\
from the point \gk A to the point \gk B\\
a \strgt, \gk{AB},\\
and there has been constructed on it\\
an equilateral triangle, \gk{DAB},\\
and suppose there have been extended\\
on a \strgt\footnotemark\ with \gk{DA} and \gk{DB}\\
the \strgt s \gk{AE} and \gk{BZ},\\
and to the center \gk B\\
at distance \gk{BG}\\
suppose a circle has been drawn,\\
\gk{GHJ},\\
and again to the center \gk D\\
at distance \gk{DH}\\
suppose a circle has been drawn, \gk{HKL}.\\
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} g`ar\\
{}>ap`o to~u A shme'iou >ep'i t`o B shme~ion\\
e>uje~ia <h AB,\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
  {}>ep> a>ut~hc\\
tr'igwnon >is'opleuron t`o DAB,\\
\kai%
\eix{\gr{ka`i}} >ekbebl'hsjwsan%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjwsan} 3rd pl perf imperat mp}\\
{}>ep> e>uje'iac ta~ic DA, DB\\
e>uje~iai a<i AE, BZ,\\
\kai%
\eix{\gr{ka`i}} k'entrw| m`en%
\eix{\gr{m'en}} t~w| B\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| BG\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o GHJ,\\
\kai%
\eix{\gr{ka`i}} p'alin\eix{\gr{p'alin}} k'entrw| t~w| D\\
\kai%
\eix{\gr{ka`i}} diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} t~w| DH\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o HKL.
}
{
\Gar\ birle\c stirilmi\c s\ olsun\\
\gk{A} noktas\i ndan \gk{B} noktas\i na\\
\gk{AB} do\u grusu,\\
ve in\c sa edilmi\c s\ olsun\\
bu [do\u gru] \"uzerine\\
e\c skenar \"u\c cgen \gk{DAB},\\
ve uzat\i lm\i\c s\ olsun\\
\gk{DA} ile \gk{DB} do\u grular\i ndan\\
\gk{AE} ile \gk{BZ} do\u grular\i,\\
ve \gk{B} merkezine\\
\gk{BG} uzakl\i \u g\i nda\\
\gk{GHJ} dairesi \c cizilmi\c s\ olsun,\\
ve \palin\ \gk{D} merkezine\\
ve \gk{DH} uzakl\i \u g\i nda\\
\gk{HKL} dairesi \c cizilmi\c s\ olsun.
}
\myfntext{The phrase \gr{>ep> e>uje'iac} will recur a number of times.  The adjective, which is feminine here, appears to be a genitive singular, though it could be accusative plural.}

\parsen{
Since then the point \gk B is the center\\
of \gk{GHJ},\\
\gk{BG} is equal to \gk{BH}.\\
Moreover,\\
since the point \gk D is the center\\
of the circle \gk{KHL},\\
equal is \gk{DL} to \gk{DH};\\
of these, the [part] \gk{DA} to \gk{DB}\\
is equal.\\
Therefore the remainder \gk{AL}\\
to the remainder \gk{BH}\\
is equal.\\
But \gk{BG} was shown equal to \gk{BH}.\\
Therefore either of \gk{AL} and \gk{BG} to \gk{BH}\\
is equal.\\
But equals to the same\\
also are equal to one another.\\
And therefore \gk{AL} is equal to \gk{BG}.
}
{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} 
t`o B shme~ion k'entron >est`i to~u GHJ,\\ 
{}>'ish >est`in <h BG t~h| BH. \\
p'alin\eix{\gr{p'alin}}, {}>epe`i%
\eix{\gr{>epe'i}} t`o D shme~ion k'entron >est`i
to~u HKL k'uklou,\\
{}>'ish >est`in <h DL t~h| DH,\\
<~wn <h DA t~h| DB {}>'ish >est'in.\\
loip`h >'ara\eix{\gr{>'ara}} <h AL\\
loip~h| t~h| BH
{}>estin >'ish.\\ 
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}} <h BG t~h| BH >'ish;\lli\\
<ekat'era >'ara\eix{\gr{>'ara}} t~wn AL, BG t~h| BH
{}>estin >'ish. \\
t`a d`e t~w| a>ut~w| >'isa\\
\kai%
\eix{\gr{ka`i}} >all'hloic >est`in >'isa; \\
\kai%
\eix{\gr{ka`i}} <h AL >'ara\eix{\gr{>'ara}} t~h| BG >estin >'ish.
}
{
\Oun\ \gk{B} noktas\i\ \gk{GHJ} dairesinin merkezi oldu\u gundan,\\
\gk{BG}, \gk{BH}'ya e\c sittir.\\
\Palin, \gk{D} noktas\i\ \gk{HKL} dairesinin merkezi oldu\u gundan,\\
\gk{DL}, \gk{DH}'ya e\c sittir,\\
ve bunlardan \gk{DA}, \gk{DB}'ya e\c sittir.\\
\Ara\ \gk{AL} kalan\i,\\
\gk{BH} kalan\i na
 e\c sittir.\\
Ve \gk{BG}'n\i n \gk{BH}'ya e\c sit oldu\u gu g\"osterilmi\c sti.\\
\Ara\ \gk{AL} ile \gk{BG}'n\i n her biri \gk{BH}'ya e\c sittir.\\
Ama ayn\i\ \c seye e\c sitler\\
birbirine de e\c sittir.\\
Ve \ara\ \gk{AL} da, \gk{BG}'ya e\c sittir.
}

\parsen{
Therefore at the given point \gk A\\
equal to the given {\strgt} \gk{BG}\\
the {\strgt} \gk{AL} is laid down;\\
\myqef
}
{
Pr`oc >'ara\eix{\gr{>'ara}} t~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iw| t~w| A\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| BG >'ish\\
e>uje~ia ke~itai%
\eix{\gr{ke~imai}!\gr{ke~itai} 3rd sg pres ind mp} <h AL;\\
<'oper >'edei poi~hsai.
}
{
\Ara\ verilmi\c s\ \gk{A} noktas\i na\\
verilmi\c s\ \gk{BG} do\u grusuna e\c sit olan\\
\gk{AL} do\u grusu oturuyor;\\
\ozqef.
}
\vfill
\begin{center}

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\uput[u](0,0){\gk D}
\uput[d](-2,-3.464){\gk E}
\uput[d](2,-3.464){\gk Z}
\uput[-15](1.5,-2.598){\gk H}
\uput[u](-0.5,0.866){\gk J}
\uput[u](-1.5,2.598){\gk K}
\uput[195](-1.5,-2.598){\gk L}
\end{pspicture}
\end{center}
\vfill
\end{proposition}



\begin{proposition}{3}%Proposition I.3

\parsen{
Two unequal {\strgt}s being given,\\
from the greater,\\
equal to the less,\\
a \strgt\ to take away.
}
{
D'uo dojeis~wn%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>ujei~wn >an'iswn\\
{}>ap`o t~hc me'izonoc\\
t~h| >el'assoni >'ishn\\
e>uje~ian >afele~in%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afele~in} aor inf act}.
}
{
\.Iki e\c sit olmayan do\u gru verilince\\
daha b\"uy\"ukten\\
daha k\"u\c c\"u\u ge e\c sit olan\\
bir do\u gru ay\i rmak.
}

\parsen{
Let be\\
the two given unequal {\strgt}s\\
\gk{AB} and \gk G,\footnotemark\\
of which let the greater be \gk{AB}. 
}
{
>'Estwsan\\
a<i doje~isai%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} d'uo e>uje~iai >'anisoi\\
a<i AB, G,\\
 <~wn me'izwn >'estw <h AB; 
}
{
Olsun\\
verilmi\c s\ iki e\c sit olmayan do\u gru\\
\gk{AB} ile \gk G,\\
ve daha b\"uy\"u\u g\"u\ \gk{AB} olsun.
}
\myfntext{Since \gk G is given the feminine gender in the Greek, this is a sign that \gk G is indeed a line and not a point.  See the Introduction.}


\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
{}>ap`o t~hc me'izonoc t~hc AB\\
t~h| >el'assoni t~h| G >'ishn\\
e>uje~ian >afele~in%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afele~in} aor inf act}.
}
{
\Deidee\\
daha b\"uy\"uk olan \gk{AB}'dan\\
daha k\"u\c c\"uk olan \gk G'ya e\c sit olan\\
bir do\u gru ay\i rmak.
}

\parsen{
Let there be laid down\\
at the point \gk A,\\
equal to the line \gk G,\\
\gk{AD};\\
and to center \gk A\\
at distance \gk{AD}\\
suppose circle \gk{DEZ} has been drawn.
}
{
Ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
pr`oc t~w| A shme'iw|\\
t~h| G e>uje'ia| >'ish
<h AD; \\
\kai%
\eix{\gr{ka`i}} k'entrw| m`en%
\eix{\gr{m'en}} t~w| A\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| AD\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o DEZ.
}
{
Otursun\\
\gk A noktas\i na\\
\gk G do\u grusuna e\c sit olan
\gk{AD}.\\
Ve \gk A merkezine\\
\gk{AD} uzakl\i \u g\i nda olan\\
\gk{DEZ} dairesi \c cizilmi\c s\ olsun.
}

\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} t`o A shme~ion\\
k'entron >est`i to~u DEZ k'uklou,\lli\\
{}>'ish >est`in <h AE t~h| AD; \\
{}>all`a%
\eix{\gr{>all'a}} \kai%
\eix{\gr{ka`i}} <h G t~h| AD >estin >'ish.\\
<ekat'era >'ara\eix{\gr{>'ara}} t~wn AE, G\\
t~h| AD >estin >'ish;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} <h AE t~h| G >estin >'ish.
}
{
Ve \gk A noktas\i,\\
\gk{DEZ} dairesinin merkezi oldu\u gundan,\\
 \gk{AE}, \gk{AD}'ya e\c sittir.  \\
Ama \gk G da, \gk{AD}'ya e\c sittir.\\
\Ara\ \gk{AE} ile \gk G'n\i n her biri\\
\gk{AD}'ya e\c sittir.\\
\Hoste\ \gk{AE} da, \gk G'ya e\c sittir.
}

\parsen{
Therefore, two unequal \strgt s being given, \gk{AB} and \gk G,\\
from the greater, \gk{AB},\\
an equal to the less, \gk G,\\
has been taken away, [namely] \gk{AE};\\
\myqef
}
{
D'uo >'ara\eix{\gr{>'ara}} dojeis~wn%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>ujei~wn >an'iswn t~wn AB, G\\
{}>ap`o t~hc me'izonoc t~hc AB\\
t~h| >el'assoni t~h| G >'ish\\
{}>af'h|rhtai%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>af'h"|rhtai} 3rd sg perf ind mp} <h AE;\\
<'oper >'edei poi~hsai.
}
{
\Ara\ iki e\c sit olmayan \gk{AB} ile \gk G do\u grusu verilince\\
daha b\"uy\"uk olan \gk{AB}'dan\\
daha k\"u\c c\"uk olan \gk G'ya e\c sit olan\\
\gk{AE} ayr\i l\i r;\\
\ozqef.
}

\begin{center}
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\uput[d](3,0){\gk B}
\uput[u](4.5,1){\gk G}
\uput[ul](-1.414,1.414){\gk D}
\uput[ur](2,0){\gk E}
\uput[d](0,-2){\gk Z}
\end{pspicture}
\end{center}
\vfill
\vfill
\end{proposition}



\begin{proposition}{4}%Proposition I.4

\parsen{
If two triangles\\
two sides\\
to two sides\\
have equal,\footnotemark\\
either [side] to either,%
\footnotemark\\ 
and angle to angle have equal,\\
---that which is by the equal {\strgt}s%
\footnotemark\\
contained,\\
also%
\footnotemark\
base to base\\
they will have equal,\\
and the triangle to the triangle\\
will be equal,\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
either to either,\\
---those that the equal sides subtend.
}
{
>E`an d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
{}[ta~ic] dus`i pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian\\
t~h| gwn'ia| >'ishn >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act} \\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp},\\
\kai%
\eix{\gr{ka`i}} t`hn b'asin\\
t`h| b'asei >'ishn <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act},\\
\kai%
\eix{\gr{ka`i}} t`o tr'igwnon\\
t~w| trig'wnw| >'ison >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i\\
<upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}}.
}
{
E\u ger iki \"u\c cgende\\
iki kenar\\
iki kenara
e\c sit olursa,\\
her biri birine,\\
ve a\c c\i,\\
a\c c\i ya e\c sit olursa,\\
{}[yani,] e\c sit do\u grular taraf\i ndan\\
i\c cerilen,\\
taban da\\
tabana e\c sit olacak,\\
\"u\c cgen de\\
\"u\c cgene e\c sit olacak,\\
ve \kalan\ a\c c\i lar da\\
\kalan\ a\c c\i lara e\c sit olacak,\\
her biri birine,\\
{}[yani,] e\c sit kenarlar taraf\i ndan\\
raptedilenler\footnotemark.
}
\footnotetext{Veya \emph{e\c sit kenarlar taraf\i ndan g\"or\"ulenler.}}

\begin{center}
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\uput[u](2.5,2.5){\gk A}
\uput[d](0,0){\gk B}
\uput[d](3,0){\gk G}
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\parsen{
Let be\\
two triangles \gk{ABG} and \gk{DEZ},\\
the two sides \gk{AB} and \gk{AG}\\
to the two sides \gk{DE} and \gk{DZ}\\
having equal,\\
 either to either,\\
\gk{AB} to \gk{DE} and \gk{AG} to \gk{DZ},\\
and angle \gk{BAG}\\
to \gk{EDZ}\\
equal.
}
{
>'Estw\\
d'uo tr'igwna t`a ABG, DEZ\\
 t`ac d'uo pleur`ac t`ac AB, AG\\
ta~ic dus`i pleura~ic ta~ic DE, DZ\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\ 
<ekat'eran <ekat'era|\\
t`hn m`en%
\eix{\gr{m'en}} AB t~h| DE t`hn d`e AG t~h| DZ \\
\kai%
\eix{\gr{ka`i}} gwn'ian t`hn <up`o BAG\\
gwn'ia| t~h| <up`o EDZ
{}>'ishn. 
}
{
Olsun\\
iki \"u\c cgen \gk{ABG} ile \gk{DEZ},\\
iki \gk{AB} ile \gk{AG} kenar\i\\
iki \gk{DE}  ile \gk{DZ} kenar\i na\\
e\c sit olan\\
her biri birine,\\
\gk{AB}, \gk{DE}'a ve \gk{AG}, \gk{DZ}'ya,\\
ve  \gk{BAG} [taraf\i ndan i\c cerilen] a\c c\i s\i\\
\gk{EDZ} a\c c\i s\i na
e\c sit [olan].
}

\parsen{
I say that\\
the base \gk{BG} is equal to the base \gk{EZ},\\
and triangle \gk{ABG}\\
will be equal to triangle \gk{DEZ},\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
either to either,\\
those that equal sides subtend,\\
{}[namely] \gk{ABG} to \gk{DEZ},\\
and \gk{AGB} to \gk{DZE}.
}
{l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} b'asic <h BG\\
b'asei t~h| EZ >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o ABG tr'igwnon\\
t~w| DEZ trig'wnw| >'ison >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i\\
<upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}},\\
<h m`en%
\eix{\gr{m'en}} <up`o ABG t~h| <up`o DEZ,\\
<h d`e <up`o AGB t~h| <up`o DZE.
}
{
Diyorum\footnotemark{} ki,\\
\gk{BG} taban\i da,\\
\gk{EZ} taban\i na e\c sittir,\\
\gk{ABG} \"u\c cgeni de\\
\gk{DEZ} \"u\c cgenine e\c sit olacak,\\
ve \kalan\ a\c c\i lar da\\
\kalan\ a\c c\i lara e\c sit olacak,\\
her biri birine,\\
e\c sit kenarlar taraf\i ndan\\
raptedilenler:\\
\gk{ABG}, \gk{DEZ}'ya,\\
ve \gk{AGB}, \gk{DZE}'a.
}\footnotetext{Veya ``\.Iddia ediyorum''.}

\parsen{
For, there being applied\\
triangle \gk{ABG}\\
to triangle \gk{DEZ},\\
and there being placed\\
the point \gk A on the point \gk D,\\
and the \strgt\ \gk{AB} on \gk{DE},\\
also the point \gk B will apply\footnotemark\ to \gk E,\\
by the equality of \gk{AB} to \gk{DE}.\\
Then, \gk{AB} applying to \gk{DE},\\
also \strgt\ \gk{AG} will apply to \gk{DZ},\\
by the equality\\
of angle \gk{BAG} to \gk{EDZ}.\\
Hence the point \gk G to the point \gk Z\\
will apply,\\
by the equality, again, of \gk{AG} to \gk{DZ}.  \\
But \gk B had applied to \gk E;\\
Hence the base \gk{BG} to the base \gk{EZ}\\
will apply.\\
For if,\\
\gk B applying to \gk E,\\
and \gk G to \gk Z,\\
the base \gk{BG} will not apply to \gk{EZ},\\
two {\strgt}s will enclose a space,\\
which is impossible.\\
Therefore will apply\\
base \gk{BG} to \gk{EZ}\\
and will be equal to it.\\
Hence triangle \gk{ABG} as a whole\\
to triangle \gk{DEZ} as a whole\\
will apply\\
and will be equal to it,\\
and the remaining angles\\
to the remaining angles\\
will apply,\\
and be equal to them,\\
\gk{ABG} to \gk{DEZ}\\
and \gk{AGB} to \gk{DZE}.
}
{
>Efarmozom'enou%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarmoz'omenoc} part pres mp} g`ar\\
to~u ABG trig'wnou\\
{}>ep`i t`o DEZ tr'igwnon\\
\kai%
\eix{\gr{ka`i}} tijem'enou%
\eix{\gr{t'ijhmi}!\gr{tij'emenoc} part pres mp}\\
to~u m`en%
\eix{\gr{m'en}} A shme'iou\\
{}>ep`i t`o D shme~ion\\
t~hc d`e AB e>uje'iac\\
{}>ep`i t`hn DE,\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act} 
\kai%
\eix{\gr{ka`i}}\\
t`o B shme~ion >ep`i t`o E\\
di`a\eix{\gr{di'a}} t`o >'ishn e>~inai t`hn AB t~h| DE;\\
{}>efarmos'ashc%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osas} part aor act} 
d`h%
\eix{\gr{d'h}}\\
t~hc AB >ep`i t`hn DE\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act} 
\kai%
\eix{\gr{ka`i}}\\
<h AG e>uje~ia >ep`i t`hn DZ \\
di`a\eix{\gr{di'a}} t`o >'ishn e>~inai
t`hn <up`o BAG gwn'ian t~h| <up`o EDZ;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} t`o G shme~ion\\
{}>ep`i t`o Z shme~ion >efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act}\\
di`a\eix{\gr{di'a}} t`o >'ishn p'alin\eix{\gr{p'alin}} e>~inai t`hn AG t~h| DZ.\\
{}>all`a%
\eix{\gr{>all'a}} m`hn%
\eix{\gr{m'hn}} \kai%
\eix{\gr{ka`i}} t`o B\\
{}>ep`i t`o E >efhrm'okei;\\
<'wste%
\eix{\gr{<'wste}} b'asic <h BG\\
{}>ep`i b'asin t`hn EZ >efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act}.\\
e>i g`ar\\
to~u m`en%
\eix{\gr{m'en}} B >ep`i t`o E >efarm'osantoc%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osas} part aor act}\\
to~u d`e G >ep`i t`o Z \\
<h BG b'asic\\
{}>ep`i t`hn EZ o>uk >efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act},\\
d'uo e>uje~iai qwr'ion peri'exousin;\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}}.\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act} >'ara\eix{\gr{>'ara}}
<h BG b'asic\\
{}>ep`i t`hn EZ\\
\kai%
\eix{\gr{ka`i}} >'ish a>ut~h| >'estai;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} <'olon t`o ABG tr'igwnon\\
{}>ep`i <'olon t`o DEZ tr'igwnon\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act}\\
\kai%
\eix{\gr{ka`i}} >'ison a>ut~w| >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
{}>ep`i t`ac loip`ac gwn'iac\\
{}>efarm'osousi%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osousi} 3rd pl fut ind act}\\
\kai%
\eix{\gr{ka`i}} >'isai a>uta~ic >'esontai,\\
<h m`en%
\eix{\gr{m'en}} <up`o ABG t~h| <up`o DEZ\\
<h d`e <up`o AGB t~h| <up`o DZE.
}
{
\Gar\ uygulan\i nca\\
\gk{ABG} \"u\c cgeni,\\
\gk{DEZ} \"u\c cgeninin \"ust\"une,\\
ve yerle\c stirilince\\
\gk A noktas\i,\\
\gk D noktas\i na,\\
ve \gk{AB} do\u grusu,\\
\gk{DE}'a,\\
uygulayacak da\\
\gk B noktas\i\ da \gk E'a,\\
\dia\ \gk{AB},  \gk{DE}'a e\c sittir.\\
\Dee\ uygulam\i\c s olunca\\ 
\gk{AB}, \gk{DE}'a,\\
uygulayacak da\\
\gk{AG} do\u grusu, \gk{DZ}'ya,\\
\dia\ \gk{BAG} a\c c\i s\i, \gk{EDZ}'ya e\c sittir.\lli\\
\Hoste\ \gk G noktas\i\ da\\
\gk Z noktas\i na uygulayacak,\\
\palin\ \dia\ \gk{AG}, \gk{DZ}'ya e\c sittir.\\
Ama \meen\ \gk B da,\\
\gk E'a uygulam\i\c st\i r;\\
\hoste\ \gk{BG} taban\i,\\
\gk{EZ} taban\i na uygulayacak.\\
\Gar\ e\u ger,\\
\gk B, \gk E'a uygulay\i nca,\\
ve \gk G, \gk Z'ya,\\
\gk{BG} taban\i\\
\gk{EZ} taban\i na uygulamayacaksa,\\
iki do\u gru bir alan i\c cerecek,\\
\imkansiz.\\
\Ara\ uygulayacak \gk{BG} taban\i,\\
\gk{EZ} taban\i na\\
ve ona e\c sit olacak.\\
Dolay\i s\i yla b\"ut\"un \gk{ABG} \"u\c cgeni de,\\
b\"ut\"un \gk{DEZ} \"u\c cgenine\\
uygulayacak,\\
ve ona e\c sit olacak,\\
ve \kalan\ a\c c\i lar\\
\kalan\ a\c c\i lara\\
uygulayacak,\\
ve onlara e\c sit olacak:\\
\gk{ABG}, \gk{DEZ}'ya\\
ve \gk{AGB}, \gk{DZE}'a.
}
\myfntext{Heath has \emph{coinciding} here, but the verb is just the active form of what, in the passive, is translated as \emph{being applied}.}

\parsen{
If, therefore, two triangles\\
two sides\\
to two sides\\
have equal,\\
either to either,\\
and angle to angle have equal,\\
---that which is by the equal {\strgt}s\\
contained,\\
also base to base\\
they will have equal,\\
and the triangle to the triangle\\
will be equal,\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
either to either,\\
---those that the equal sides subtend;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
{}[ta~ic] d'uo pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\ 
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian\\
t~h| gwn'ia| >'ishn >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp},\\
\kai%
\eix{\gr{ka`i}} t`hn b'asin\\
t`h| b'asei >'ishn <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act},\\
\kai%
\eix{\gr{ka`i}} t`o tr'igwnon\\
t~w| trig'wnw| >'ison >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i\\
<upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
<'oper >'edei de~ixai.
}
{
\Ara,  e\u ger iki \"u\c cgende\\
iki kenar\\
iki kenara e\c sit olursa\\
(her biri birine)\\
ve a\c c\i\\
a\c c\i ya e\c sit olursa\\
{}[yani,] e\c sit do\u grular taraf\i ndan\\
i\c cerilen,\\
taban da\\
tabana e\c sit olacak,\\
\"u\c cgen de\\
\"u\c cgene e\c sit olacak,\\
ve \kalan\ a\c c\i lar da\\
\kalan\ a\c c\i lara e\c sit olacak,\\
her biri birine,\\
{}[yani] e\c sit kenarlar taraf\i ndan\\
raptedilenler;\\
\ozqed.
}

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\end{proposition}

%\newpage

\begin{proposition}{5}%Proposition I.5

\parsen{
In\footnotemark\ isosceles triangles,\\
the angles at the base\\
are equal to one another,\\
and,\\
the equal \strgt s being extended,\\
the angles under the base\\
will be equal to one another.
}
{T~wn >isoskel~wn trig'wnwn\\
a<i pr`oc t~h| b'asei gwn'iai\\
{}>'isai >all'hlaic e>is'in,
\kai%
\eix{\gr{ka`i}}\\
prosekblhjeis~wn%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass} t~wn >'iswn e>ujei~wn\\
a<i <up`o t`hn b'asin gwn'iai\\
{}>'isai >all'hlaic >'esontai.
}
{
\.Ikizkenar \"u\c cgenlerde,\\
tabandaki a\c c\i lar\\
birbirine e\c sittir,
ve,\\
e\c sit do\u grular uzat\i ld\i \u g\i nda,\\
taban\i n alt\i nda \kalan\ a\c c\i lar\\
birbirine e\c sit olacak.\\
}


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\parsen{
Let there be\\
an isosceles triangle, \gk{ABG}\\
having equal\\
side \gk{AB} to side \gk{AG},\\ 
and suppose have been extended\\
on a \strgt\ with \gk{AB} and \gk{AG}\\
the {\strgt}s \gk{BD} and \gk{GE}.
}
{
>'Estw\\
tr'igwnon >isoskel`ec t`o ABG\\
{}>'ishn >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}
t`hn AB pleur`an t~h| AG pleur~a|,\\ 
\kai%
\eix{\gr{ka`i}} prosekbebl'hsjwsan%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekbebl'hsjwsan} 3rd pl perf imperat mp}\\
{}>ep> e>uje'iac ta~ic AB, AG\\
e>uje~iai a<i BD, GE; 
}
{
Olsun\\
ikizkenar \"u\c cgen \gk{ABG},\\
\gk{AB} kenar\i\ \gk{AG} kenar\i na e\c sit olan,\lli\\ 
ve uzat\i lm\i\c s olsun\\
\gk{AB} ve \gk{AG} do\u grular\i ndan\\
\gk{BD} ve \gk{GE} do\u grular\i.  
}

\parsen{
I say that\\
angle \gk{ABG} to angle \gk{AGB}\\
is equal,\\
and \gk{GBD} to \gk{BGE}.
}
{l'egw, <'oti\\
<h m`en%
\eix{\gr{m'en}} <up`o ABG gwn'ia\\
t~h| <up`o AGB {}>'ish >est'in,\\
<h d`e <up`o GBD t~h| <up`o BGE.
}
{
\Legohoti\\
\gk{ABG} a\c c\i s\i,\\
\gk{AGB}'ya e\c sittir\\
ve \gk{GBD}, \gk{BGE}'a e\c sittir 
}

\parsen{
For, suppose there has been chosen\\
a random point \gk Z on \gk{BD},\\
and there has been taken away\\
from the greater, \gk{AE},\\
to the less, \gk{AZ},\\
an equal, \gk{AH},\\
and suppose there have been joined\\
the {\strgt}s \gk{ZG} and \gk{HB}.
}
{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp}%
\glossary{\gr{e>il'hfjw}}
g`ar\\
{}>ep`i t~hc BD\\
tuq`on shme~ion t`o Z,\\
\kai%
\eix{\gr{ka`i}} >afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}\\
{}>ap`o t~hc me'izonoc t~hc AE\\
t~h| >el'assoni t~h| AZ {}>'ish <h AH,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp}\\
a<i ZG, HB e>uje~iai.
}
{
\Gar\ al\i nm\i\c s olsun\\
\gk{BD} \"uzerinde\\
rastgele bir \gk Z noktas\i,\\ 
ve ayr\i lm\i\c s olsun\\
b\"uy\"uk olan \gk{AE}'dan\\
k\"u\c c\"uk olan \gk{AZ}'ya e\c sit olan \gk{AH},\\
ve birle\c stirilmi\c s\ olsun\\
\gk{ZG} ve \gk{HB} do\u grular\i.
}

\parsen{
Since then \gk{AZ} is equal to \gk{AH},\\
and \gk{AB} to \gk{AG},\\
so the two \gk{AZ} and \gk{AG}\\
to the two \gk{HA}, \gk{AB},\\
will be equal,\\
either to either;\\
and they bound a common angle,\\
{}[namely] \gk{ZAH};\\
therefore the base \gk{ZG} to the base \gk{HB}\\
is equal,\\
and triangle \gk{AZG} to triangle \gk{AHB}\\
will be equal,\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
either to either,\\
those that the equal sides subtend,\\
\gk{AGZ} to \gk{ABH},\\
and \gk{AZG} to \gk{AHB}.\\
And since \gk{AZ} as a whole\\
to \gk{AH} as a whole\\
is equal,\\
of which the [part] \gk{AB} to \gk{AG} is equal,\\
therefore the remainder \gk{BZ}\\
to the remainder \gk{GH}\\
is equal.\\
And \gk{ZG} was shown equal to \gk{HB}.\\
Then the two \gk{BZ} and \gk{ZG}\\
to the two \gk{GH} and \gk{HB}\\
are equal,\\
either to either,\\
and angle \gk{BZG}\\
to angle \gk{GHB}\\
{}[is] equal,\\
and the common base of them is \gk{BG};\\
and therefore triangle \gk{BZG}\\
to triangle \gk{GHB}\\
will be equal,\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
either to either,\\
which the equal sides subtend.\\
Equal therefore is\\
\gk{ZBG} to \gk{HGB},\\
and \gk{BGZ} to \gk{GBH}.\\
Since then angle \gk{ABH} as a whole\\
to angle \gk{AGZ} as a whole\\
was shown equal,\\
of which the [part] \gk{GBH} to \gk{BGZ}\\
is equal,\\
therefore the remainder \gk{ABG}\\
to the remainder \gk{AGB}\\
is equal;\\
and they are at the base\\
of the triangle \gk{ABG}.\\
And was shown also\\
\gk{ZBG} equal to \gk{HGB};\\
and they are under the base.
}
{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} AZ t~h| AH\\
<h d`e AB t~h| AG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i ZA, AG\\
dus`i ta~ic HA, AB
{}>'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ian koin`hn peri'eqousi%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act}\\
t`hn <up`o ZAH;\\
b'asic >'ara\eix{\gr{>'ara}} <h ZG b'asei\\
t~h| HB {}>'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o AZG tr'igwnon\\
t~w| AHB trig'wnw| {}>'ison >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic
{}>'isai >'esontai\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}},\\
<h m`en%
\eix{\gr{m'en}} <up`o AGZ t~h| <up`o ABH,\\
<h d`e <up`o AZG t~h| <up`o AHB.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} <'olh <h AZ\\
<'olh| t~h| AH
{}>estin >'ish,\\
<~wn <h AB\\
t~h| AG >estin >'ish,\\
loip`h >'ara\eix{\gr{>'ara}} <h BZ\\
loip~h| t~h| GH
{}>estin >'ish.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}} <h ZG\\
t~h| HB >'ish;\\
d'uo d`h%
\eix{\gr{d'h}} a<i BZ, ZG\\
dus`i ta~ic GH, HB {}>'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o BZG\\
 gwn'ia| th| <up`o GHB {}>'ish,\\
\kai%
\eix{\gr{ka`i}} b'asic a>ut~wn koin`h <h BG;\\
\kai%
\eix{\gr{ka`i}} t`o BZG >'ara\eix{\gr{>'ara}} tr'igwnon\\
t~w| GHB trig'wnw|
{}>'ison >'estai,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic
{}>'isai >'esontai\\
<ekat'era  <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
{}>'ish >'ara\eix{\gr{>'ara}} >est`in\\
<h m`en%
\eix{\gr{m'en}} <up`o ZBG t~h| <up`o HGB\\
<h d`e <up`o BGZ t~h| <up`o GBH.\\
{}>epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} <'olh <h <up`o ABH gwn'ia\\
<'olh| t~h| <up`o AGZ gwn'ia|\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} >'ish,\\
<~wn <h <up`o GBH\\
t~h| <up`o BGZ {}>'ish,\\
loip`h >'ara\eix{\gr{>'ara}} <h <up`o ABG\\
loip~h| t~h| <up`o AGB
{}>estin >'ish;\\
ka'i e>isi pr`oc t~h| b'asei\\
to~u ABG trig'wnou.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}}
<h <up`o ZBG\\
t~h| <up`o HGB >'ish;\\
ka'i e>isin <up`o t`hn b'asin.
}
{
\Oun\ e\c sit oldu\u gundan\\
\gk{AZ}, \gk{AH}'ya\\
ve \gk{AB}, \gk{AG}'ya,\\
\dee\ \gk{ZA}, \gk{AG} ikilisi\\ 
\gk{HA}, \gk{AB} ikilisine e\c sittir,\\
her biri birine;\\
ve ortak bir a\c c\i y\i\ s\i n\i rland\i r\i rlar,\\
(yani) \gk{ZAH}'y\i;\\
\ara\ \gk{ZG} taban\i\\
\gk{HB} taban\i na e\c sittir,\\
ve \gk{AZG} \"u\c cgeni\\
\gk{AHB} \"u\c cgenine e\c sit olacak,\\
ve \kalan\ a\c c\i lar\\
\kalan\ a\c c\i lara e\c sit olacak,\\
her biri birine,\\
(yani) e\c sit kenarlar\i\ g\"orenler;\\
\gk{AGZ}, \gk{ABH}'ya,\\
ve \gk{AZG}, \gk{AHB}'ya.\\
Ve b\"ut\"un \gk{AZ}\\
b\"ut\"un \gk{AH}'ya e\c sit oldu\u gundan,\\
ve bunlar\i n [par\c calar\i ndan] \gk{AB}\\
\gk{AG}'ya e\c sit oldu\u gundan,\\
\ara\ \gk{BZ} kalan\i\\
\gk{GH} kalan\i na e\c sittir.\\
Ve g\"osterilmi\c sti \gk{ZG}'n\i n\\
\gk{HB}'ya e\c sit oldu\u gu.\\
\Dee\ \gk{BZ} ve \gk{ZG} ikilisi\\
\gk{GH} ve \gk{HB} ikilisine e\c sittir,\\
her biri birine,\\
ve \gk{BZG} a\c c\i s\i,\\
\gk{GHB} a\c c\i s\i na e\c sittir,\\
ve onlar\i n ortak taban\i\ \gk{BG}'d\i r;\\
\Ara\ \gk{BZG} \"u\c cgeni de\\
\gk{GHB} \"u\c cgenine e\c sit olacak,\\
ve \kalan\ a\c c\i lar\\
\kalan\ a\c c\i lar\i na e\c sit olacak,\\
her biri birine,\\
ayn\i\ kenarlar\i\ g\"orenler.\\
\Ara\ e\c sittir\\
\gk{ZBG}, \gk{HGB}'ya,\\
ve \gk{BGZ}, \gk{GBH}'ya.\\
\Oun\ b\"ut\"un \gk{ABH} a\c c\i s\i n\i n\\
b\"ut\"un \gk{AGZ} a\c c\i s\i na\\
e\c sit oldu\u gu g\"osterilmi\c s oldu\u gundan\\
ve bunlar\i n [par\c calar\i ndan] \gk{GBH},\\
\gk{BGZ}'ya e\c sit oldu\u gundan,\\
\ara\ kalan \gk{ABG},\\
kalan \gk{AGB}'ya e\c sittir;\\
ve bunlar taban\i ndad\i r\\
\gk{ABG} \"u\c cgeninin.\\
Ve g\"osterilmi\c sti \gk{ZBG}'n\i n\\
\gk{HGB}'ya e\c sit oldu\u gu;\\
ve bunlar taban\i n alt\i ndad\i r.
}

\parsen{
Therefore, in isosceles triangles,\\
the angles at the base\\
are equal to one another,\\
and,\\
the equal \strgt s being extended,\\
the angles under the base\\
will be equal to one another;\\
\myqed
}
{T~wn >'ara\eix{\gr{>'ara}} >isoskel~wn trig'wnwn\\
a<i pr`oc t~h| b'asei gwn'iai\\
{}>'isai >all'hlaic e>is'in,
\kai%
\eix{\gr{ka`i}}\\
prosekblhjeis~wn%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass} t~wn >'iswn e>ujei~wn\\
a<i <up`o t`hn b'asin gwn'iai\\
{}>'isai >all'hlaic >'esontai;\\
<'oper >'edei de~ixai.
}
{
\Ara\ ikizkenar \"u\c cgenlerde,\\
tabandaki a\c c\i lar\\
birbirine e\c sittir,
ve,\\
e\c sit do\u grular uzat\i ld\i \u g\i nda,\\
taban\i n alt\i nda \kalan\ a\c c\i lar\\
birbirine e\c sit olacak;\\
\ozqed.
}

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\end{proposition}



\begin{proposition}{6}%Proposition I.6

\parsen{
If in a triangle\\
two angles be equal to one another,\\
also the sides that subtend the equal angles\\
will be equal to one another.
}
{>E`an trig'wnou
a<i d'uo gwn'iai\\
{}>'isai >all'hlaic >~wsin,\\
\kai%
\eix{\gr{ka`i}} a<i <up`o t`ac >'isac gwn'iac\\
<upote'inousai%
\eix{\gr{te'inw}!\gr{<upote'inw}!\gr{<upote'inwn} part pres act} pleura`i\\
{}>'isai >all'hlaic >'esontai.}
{
E\u ger bir \"u\c cgenin iki a\c c\i s\i\\
birbirine e\c sit ise,\\
e\c sit a\c c\i lar\i\\
rapteden kenarlar da\\
birbirine e\c sit olacakt\i r.}

\parsen{
Let there be\\
a triangle, \gk{ABG},\\
having equal\\
angle \gk{ABG}\\
to angle \gk{AGB}.\\
}
{
>'Estw\\
tr'igwnon t`o ABG\\
{}>'ishn >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}
t`hn <up`o ABG gwn'ian\\
t~h| <up`o AGB gwn'ia|;
}
{
Olsun\\
\"u\c cgen \gk{ABG},\\
\gk{ABG} a\c c\i s\i\ e\c sit olan\\
 \gk{AGB} a\c c\i s\i na.\\
}


\parsen{
I say that\\
also side \gk{AB} to side \gk{AG}\\
is equal.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} pleur`a <h AB\\
pleur~a| t~h| AG {}>estin >'ish.
}
{
\Legohoti\\
\gk{AB} kenar\i\ da\\
\gk{AG} kenar\i na e\c sittir.
}

\parsen{
For if unequal is \gk{AB} to \gk{AG},\\
one of them is greater.\\
Suppose \gk{AB} be greater,\\
and there has been taken away\\
from the greater, \gk{AB},\\
to the less, \gk{AG},\\
an equal, \gk{DB},\\
and there has been joined \gk{DG}.
}
{
E>i g`ar >'anis'oc >estin <h AB t~h| AG,\\
<h <et'era a>ut~wn me'izwn >est'in.\\
{}>'estw me'izwn <h AB,\\
\kai%
\eix{\gr{ka`i}} >afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}\\
{}>ap`o t~hc me'izonoc t~hc AB\\
t~h| >el'attoni t~h| AG {}>'ish\\
<h DB,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h DG.
}
{
\Gar\ e\u ger \gk{AB}, \gk{AG}'ya e\c sit de\u gilse,\\
biri daha b\"uy\"ukt\"ur.\\
\gk{AB} daha b\"uy\"uk olsun,\\
ve ayr\i lm\i\c s olsun\\
daha b\"uy\"uk olan \gk{AB}'dan\\
daha k\"u\c c\"uk olan \gk{AG}'ya e\c sit olan\\
\gk{DB},\\
ve \gk{DG} birle\c stirilmi\c s\ olsun.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in <h DB t~h| AG\lli\\
koin`h d`e <h BG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i DB, BG\\
d'uo ta~ic AG, GB
{}>'isai e>is`in\\
<ekat'era <ekat'era|,\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o DBG\\
gwn'ia| t~h| <up`o AGB
{}>estin >'ish;\\
b'asic >'ara\eix{\gr{>'ara}} <h DG\\
b'asei t~h| AB {}>'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o DBG tr'igwnon\\
t~w| AGB trig'wnw| {}>'ison >'estai,\\
t`o >'elasson t~w| me'izoni;\\
<'oper >'atopon;\\
o>uk >'ara\eix{\gr{>'ara}} >'anis'oc >estin\\
<h AB t~h| AG;\\
{}>'ish >'ara\eix{\gr{>'ara}}.
}
{
\Oun\ \gk{DB}, \gk{AG}'ya e\c sit oldu\u gundan,\\
ve \gk{BG} ortak oldu\u gundan,\\
\dee\ \gk{DB}, \gk{BG} ikilisi\\
\gk{AG}, \gk{BG} ikilisine  e\c sittir\\
her biri birine,\\
ve \gk{DBG} a\c c\i s\i\\
\gk{AGB} a\c c\i s\i na e\c sittir;\\
\ara\  \gk{DG} taban\i\\
\gk{AB} taban\i na e\c sittir,\\
ve \gk{DBG} \"u\c cgeni\\
\gk{AGB} \"u\c cgenine e\c sit olacak,\\
daha k\"u\c c\"uk daha b\"uy\"u\u ge;\\
\sacma;\\
\ara\ e\c sit de\u gil de\u gildir\\
\gk{AB}, \gk{AG}'ya;\\
\ara\ e\c sittir.
}

\parsen{
If therefore in a triangle\\
two angles be equal to one another,\\
also the sides that subtend the equal angles\\
will be equal to one another;\\
\myqed
}
{>E`an >'ara\eix{\gr{>'ara}} trig'wnou
a<i d'uo gwn'iai\\
{}>'isai >all'hlaic >~wsin,\\
\kai%
\eix{\gr{ka`i}} a<i <up`o t`ac >'isac gwn'iac\\
<upote'inousai%
\eix{\gr{te'inw}!\gr{<upote'inw}!\gr{<upote'inwn} part pres act} 
pleura`i\\
{}>'isai >all'hlaic >'esontai;\\
<'oper >'edei de~ixai.
}
{
\Ara\ e\u ger bir \"u\c cgenin iki a\c c\i s\i\\
birbirine e\c sit ise,\\
e\c sit a\c c\i lar\i\\
rapteden kenarlar da\\
birbirine e\c sit olacaklar;\\
\ozqed.
}

\begin{center}
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} 

\end{center}


\end{proposition}


	
\begin{proposition}{7}%Proposition I.7

\parsen{
On the same \strgt,\\
to the same two \strgt s,\\
two other \strgt s,\\
{}[which are] equal,\\
either to either,\\
will not be constructed\\
to one and another point,\footnotemark\\
to the same parts,\footnotemark\\
having the same extremities\\
as\footnotemark\ the original lines.}
{
>Ep`i t~hc a>ut~hc e>uje'iac\\
d'uo ta~ic a>uta~ic e>uje'iaic\\
{}>'allai d'uo e>uje~iai {}>'isai\\
<ekat'era <ekat'era|\\
o>u sustaj'hsontai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaj'hsontai} 3rd pl fut ind pass}\\
pr`oc >'allw| \kai%
\eix{\gr{ka`i}} >'allw| shme'iw|\\
{}>ep`i t`a a>ut`a m'erh\\
t`a a>ut`a p'erata >'eqousai%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
ta~ic >ex >arq~hc e>uje'iaic.
}
{
Ayn\i\ do\u gru \"uzerinde,\\
ayn\i\ iki do\u gruya\\
e\c sit olan ba\c ska iki do\u gru,\\
her biri birine,\\
in\c sa edilmeyecek\\
bir ve ba\c ska bir noktaya,\\
ayn\i\ tarafta,\\
ayn\i\ s\i n\i rlar\i\ olan\\
ba\c slang\i\c ctaki do\u grularla.\footnotemark
}\footnotetext{Heath \cite[I.259]{MR17:814b}, 21.\ \"onermeyle kar\c s\i la\c st\i rmam\i z\i\ \"onerir.}

\parsen{
For if possible,\\
on the same {\strgt} \gk{AB}\\
to two given {\strgt}s \gk{AG}, \gk{GB},\\
two other {\strgt}s \gk{AD}, \gk{DB},\\
equal\\
either to either\\
suppose have been constructed\footnotemark\\
to one and another point\\
\gk G and \gk D,\\
to the same parts,\\
having the same extremities,\\
so that \gk{GA} is\footnotemark\ equal to \gk{DA},\\
having the same extremity as it, \gk A,\\
and \gk{GB} to \gk{DB},\\
having the same extremity as it, \gk B,\\
and suppose there has been joined\\
\gk{GD}.
}
{
E>i g`ar dunat'on,\\
{}>ep`i t~hc a>ut~hc e>uje'iac t~hc AB\\
d'uo ta~ic a>uta~ic e>uje'iaic ta~ic AG, GB\\
{}>'allai d'uo e>uje~iai a<i AD, DB {}>'isai\\
<ekat'era <ekat'era|\\
sunest'atwsan%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atwsan} 3rd pl perf
  imperat mp}\\
pr`oc >'allw| \kai%
\eix{\gr{ka`i}} >'allw| shme'iw|\\
t~w| te%
\eix{\gr{te}} G \kai%
\eix{\gr{ka`i}} D\\
{}>ep`i t`a a>ut`a m'erh\\
t`a a>ut`a p'erata >'eqousai%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act},\\
<'wste%
\eix{\gr{<'wste}} >'ishn e>~inai\\
t`hn m`en%
\eix{\gr{m'en}} GA t~h| DA\\
t`o a>ut`o p'erac >'eqousan%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} a>ut~h|\\
t`o A,\\
t`hn d`e GB t~h| DB\\
t`o a>ut`o p'erac >'eqousan%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} a>ut~h|\\
t`o B,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h GD.
}
{
\Gar\ e\u ger m\"umk\"unse,\\
ayn\i\ \gk{AB} do\u grusu \"uzerinde\\
verilmi\c s\ iki \gk{AG}, \gk{GB} do\u grusuna\\
e\c sit ba\c ska iki \gk{AD}, \gk{DB} do\u grusu\\
her biri birine\\
in\c sa edilmi\c s\ olsun\\
bir ve ba\c ska bir noktaya,\\
hem \gk G'ya hem \gk D'ya,\\
ayn\i\ tarafta,\\
ayn\i\ s\i n\i rlar\i\ olan,\\
\hosteB\ e\c sit olsun\\
hem \gk{GA}, \gk{DA}'ya,\\
kendisiyle ayn\i\ s\i n\i ra sahip olan,\\
{}[yani] \gk A;\\
hem de \gk{GB}, \gk{DB}'ya,\\
kendisiyle ayn\i\ s\i n\i ra sahip olan,\\
{}[yani] \gk B,\\
ve \gk{GD} birle\c stirilmi\c s\ olsun.
}

\parsen{
Because equal is \gk{AG} to \gk{AD},\\
equal is\\
also angle \gk{AGD} to \gk{ADG};\\
Greater therefore [is]\\
\gk{ADG} than\footnotemark\ \gk{DGB};\footnotemark\\
by much, therefore, [is]\\
\gk{GDB} greater than \gk{DGB}.\\
Moreover, since equal is \gk{GB} to \gk{DB},\\
equal is also\\
angle \gk{GDB} to angle \gk{DGB}.\\
But it was also shown than it\\
much greater;\\
which is absurd.
}
{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h AG t~h|  AD,\\
{}>'ish >est`i\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o AGD t~h| <up`o ADG;\\
me'izwn >'ara\eix{\gr{>'ara}} <h <up`o ADG\\
t~hc <up`o DGB;\\
poll~w| >'ara\eix{\gr{>'ara}} <h <up`o GDB me'izwn >est'i\\
t~hc <up`o DGB.\\
p'alin\eix{\gr{p'alin}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in <h GB t~h| DB,\\
{}>'ish >est`i\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o GDB\\
gwn'ia| t~h| <up`o DGB.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e a>ut~hc \kai%
\eix{\gr{ka`i}} poll~w| me'izwn;\lli\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}}.
}
{
\Oun\ e\c sit oldu\u gundan\\
\gk{AG}, \gk{AD}'ya,\\
e\c sittir\\
\gk{AGD} a\c c\i s\i\ da, \gk{ADG}'ya;\\
\ara\ \gk{ADG} b\"uy\"ukt\"ur\\
\gk{DGB}'dan;\\
\ara\ \gk{GDB} \c cok daha b\"uy\"ukt\"ur\\
\gk{DGB}'dan.\\
\Palin\ \gk{GB}, \gk{DB}'ya e\c sit oldu\u gundan,\\
e\c sittir\\
\gk{GDB} a\c c\i s\i\ da,\\
\gk{DGB} a\c c\i s\i na.\\
Ve ondan \c cok daha b\"uy\"uk oldu\u gu g\"osterilmi\c sti;\\
\imkansiz.
}

\parsen{
Not, therefore,\\
on the same {\strgt},\\
to the same two {\strgt}s,\\
two other {\strgt}s\\
{}[which are] equal,\\
either to either,\\
will be constructed\\
to one and another point\\
to the same parts\\
having the same extremities\\
as the original lines;\\
\myqed
}
{
O>uk  >'ara\eix{\gr{>'ara}}\\
{}>ep`i t~hc a>ut~hc e>uje'iac\\
d'uo ta~ic a>uta~ic e>uje'iaic\\
{}>'allai d'uo e>uje~iai {}>'isai\\
<ekat'era <ekat'era|\\
sustaj'hsontai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaj'hsontai} 3rd pl fut ind pass}\\
pr`oc >'allw| \kai%
\eix{\gr{ka`i}} >'allw| shme'iw|\\
{}>ep`i t`a a>ut`a m'erh\\
t`a a>ut`a p'erata >'eqousai%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
ta~ic >ex >arq~hc e>uje'iaic;\\
<'oper >'edei de~ixai.
}
{
\Ara\ olmaz:\\
ayn\i\ do\u gru \"uzerinde,\\
iki verilmi\c s do\u gruya,\\
e\c sit iki ba\c ska do\u gru,\\
her biri birine,\\
in\c sa edilmeyecek\\
bir ve ba\c ska bir noktaya,\\
ayn\i\ tarafta,\\
ayn\i\ s\i n\i rlar\i\ olan\\
ba\c slang\i\c ctaki do\u grularla;\\
\ozqed.
}

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\end{center}

\end{proposition}





\begin{proposition}{8}%Proposition I.8

\parsen{
If two triangles\\
two sides\\
to two sides\\
have equal,\\
either to either,\\
and have also base equal to base,\\
also angle to angle\\
they will have equal,\\
{}[namely] that by the equal \strgt s\\
subtended.
}
{
>E`an d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
{}[ta~ic] d'uo pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|,\\
{}>'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act} 
d`e \kai%
\eix{\gr{ka`i}} t`hn b'asin t~h| b'asei >'ishn,\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian t~h| gwn'ia|
{}>'ishn <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp}.
}
{
E\u ger iki \"u\c cgende\\
iki kenar\\
iki kenara e\c sit ise,\\
her biri birine,\\
ve taban tabana  e\c sit ise,\\ 
a\c c\i\ da a\c c\i ya e\c sit olacak,\\
e\c sit do\u grularca\\
i\c cerilen.
}

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\end{center}

\parsen{
Let there be\\
two triangles, \gk{ABG} and \gk{DEZ},\\
the two sides \gk{AB} and \gk{AG}\\
to the two sides \gk{DE} and \gk{DZ}\\
having equal,\\
either to either,\\
\gk{AB} to \gk{DE},\\
and \gk{AG} to \gk{DZ};\\
and let them have\\
base \gk{BG} equal to base \gk{EZ}.
}
{
>'Estw\\
d'uo tr'igwna t`a ABG, DEZ\\
t`ac d'uo pleur`ac t`ac AB, AG\\
ta~ic d'uo pleura~ic ta~ic DE, DZ\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
<ekat'eran <ekat'era|,\\
t`hn m`en%
\eix{\gr{m'en}} AB t~h| DE\\
t`hn d`e AG t~h| DZ;\\
{}>eq'etw d`e\\
\kai%
\eix{\gr{ka`i}} b'asin t`hn BG b'asei t~h| EZ >'ishn;
}
{
Olsun\\
iki \"u\c cgen \gk{ABG} ve \gk{DEZ},\\
iki \gk{AB} ile \gk{AG} kenar\i\\
iki \gk{DE} ile \gk{DZ} kenar\i na\\
e\c sit olan\\
her biri birine,\\
\gk{AB}, \gk{DE}'a,\\
\gk{AG} da, \gk{DZ}'ya;\\
olsun\\
bir de \gk{BG} taban\i\ \gk{EZ} taban\i na e\c sit.
}

\parsen{
I say that\\
also angle \gk{BAG}\\
to angle \gk{EDZ}\\
is equal.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o BAG\\
gwn'ia| t~h| <up`o EDZ {}>estin >'ish.
}
{
\Legohoti\\
bir de \gk{BAG} a\c c\i s\i\ da\\
\gk{EDZ} a\c c\i s\i na e\c sittir.
}

\parsen{
For, there being applied\\
triangle \gk{ABG}\\
to triangle \gk{DEZ},\\
and there being placed\\
the point \gk B on the point \gk E,\\
and the \strgt{} \gk{BG} on \gk{EZ},\\
also the point \gk G will apply to \gk Z,\\
by the equality of \gk{BG} to \gk{EZ}.\\
Then, \gk{BG} applying to \gk{EZ},\\
also will apply\\
\gk{BA} and \gk{GA} to \gk{ED} and \gk{DZ}.\\
For if base \gk{BG} to the base \gk{EZ}\\
apply,\\
and sides \gk{BA}, \gk{AG} to \gk{ED}, \gk{DZ}\\
do not apply,\\
but deviate,\\
as \gk{EH}, \gk{HZ},\\
there will be constructed\\
on the same \strgt,\\
to two given \strgt s,\\
two other \strgt s equal,\\
either to either,\\
to one and another point\\
to the same parts\\
having the same extremities.\\
But they are not constructed;\\
therefore it is not [the case] that,\\
there being applied\\
the base \gk{BG} to the base \gk{EZ},\\
there do not apply\\
sides \gk{BA}, \gk{AG} to \gk{ED}, \gk{DZ}.\\
Therefore they apply.\\
So angle \gk{BAG}\\
to angle \gk{EDZ}\\
will apply\\
and will be equal to it.
}
{
>Efarmozom'enou%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarmoz'omenoc} part pres mp} g`ar\\
to~u ABG trig'wnou\\
{}>ep`i t`o DEZ tr'igwnon\\
\kai%
\eix{\gr{ka`i}} tijem'enou%
\eix{\gr{t'ijhmi}!\gr{tij'emenoc} part pres mp}\\
to~u m`en%
\eix{\gr{m'en}} B shme'iou >ep`i t`o E shme~ion\\
t~hc d`e BG e>uje'iac >ep`i t`hn EZ\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act} \kai%
\eix{\gr{ka`i}}\\
t`o G shme~ion >ep`i t`o Z\\
di`a\eix{\gr{di'a}} t`o >'ishn e>~inai t`hn BG t~h| EZ;\\
{}>efarmos'ashc%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osas} part aor act} d`h%
\eix{\gr{d'h}}\\
t~hc BG >ep`i t`hn EZ\\
{}>efarm'osousi%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osousi} 3rd pl fut ind act} \kai%
\eix{\gr{ka`i}} a<i BA, GA\\
{}>ep`i t`ac ED, DZ.\\
e>i g`ar b'asic m`en%
\eix{\gr{m'en}} <h BG\\
{}>ep`i b'asin t`hn EZ >efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act},\\
a<i d`e BA, AG pleura`i\\
{}>ep`i t`ac ED, DZ o>uk >efarm'osousin%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osousi} 3rd pl fut ind act}\\
{}>all`a%
\eix{\gr{>all'a}} parall'axousin%
\eix{\gr{>all'attw}!\gr{parall'attw}!\gr{parall'axousi} 3rd pl fut ind act}\\
<wc a<i EH, HZ,\\
sustaj'hsontai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaj'hsontai} 3rd pl fut ind pass}\\
{}>ep`i t~hc a>ut~hc e>uje'iac\\
d'uo ta~ic a>uta~ic e>uje'iaic\\
{}>'allai d'uo e>uje~iai >'isai\\
<ekat'era <ekat'era|\\
pr`oc >'allw| \kai%
\eix{\gr{ka`i}} >'allw| shme'iw|\\
{}>ep`i t`a a>ut`a m'erh\\
t`a a>ut`a p'erata >'eqousai%
\eix{\gr{>'eqw}}.\\
o>u sun'istantai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'istantai} 3rd pl pres ind mp} d'e;\\
o>uk >'ara\eix{\gr{>'ara}}\\
{}>efarmozom'enhc%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarmoz'omenoc} part pres mp}
t~hc BG b'asewc\\
{}>ep`i t`hn EZ b'asin\\
o>uk >efarm'osousi%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osousi} 3rd pl fut ind act}\\
\kai%
\eix{\gr{ka`i}} a<i BA, AG pleura`i\\
{}>ep`i t`ac ED, DZ.\\
{}>efarm'osousin%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osousi} 3rd pl fut ind act} >'ara\eix{\gr{>'ara}};\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o BAG\\
{}>ep`i  gwn'ian t`hn <up`o EDZ\\
{}>efarm'osei%
\eix{\gr{<arm'ozw}!\gr{>efarm'ozw}!\gr{>efarm'osei} 3rd sg fut ind act}\\
\kai%
\eix{\gr{ka`i}} >'ish a>ut~h| >'estai.
}
{
\Gar\ uygulan\i nca\\
 \gk{ABG} \"u\c cgeni\\
\gk{DEZ} \"u\c cgene,\\
ve yerle\c stirilince\\
\gk B noktas\i, \gk E noktas\i na,\\
ve \gk{BG} do\u grusu, \gk{EZ}'ya,\\
uygulayacak da\\
\gk G noktas\i, \gk Z'ya,\\
\dia\ \gk{BG}, \gk{EZ}'ya e\c sittir.\\
Uygulay\i nca, \dee,\\
\gk{BG}, \gk{EZ}'ya,\\
bir de \gk{BA} ve \gk{GA}, uygulayacak\\
\gk{ED} ve \gk{DZ}'ya.\\
\Gar\ e\u ger  \gk{BG} taban\i,\\ 
\gk{EZ} taban\i na uygularsa,\\
ve \gk{BA} ve \gk{AG} kenarlar\i\\
\gk{ED} ve \gk{DZ}'ya uygulamazsa,\\
ama saparsa,\\
\gk{EH} ve \gk{HZ} olarak,\\
in\c sa edilecek\\
ayn\i\ do\u gru \"uzerinde,\\
ayn\i\ iki do\u gruya\\
e\c sit olan ba\c ska iki do\u gru,\\
her biri birine,\\
bir ve ba\c ska bir noktaya\\
ayn\i\ tarafta\\
ayn\i\ s\i n\i rlar\i\ olan.\\
Ama in\c sa edilmez;\\
\ara\ olmaz:\\
\gk{BG} taban\i\ uygulay\i nca\\
\gk{EZ} taban\i na,\\
uygulamayacak\\
\gk{BA} ve \gk{AG} kenarlar\i\ da,\\
\gk{ED} ve \gk{DZ}'ya.\\
\Ara\ uygulayacaklar.\\
\Hoste\ \gk{BAG} a\c c\i s\i\ da\\
\gk{EDZ} a\c c\i s\i na\\
uygulayacak\\
ve ona e\c sit olacak.
}


\parsen{
If, therefore, two triangles\\
two sides\\
to two sides\\
have equal,\\
either to either,\\
and have also base equal to base,\\
also angle to angle\\
they will have equal,\\
{}[namely] that by the equal \strgt s\\
subtended;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
{}[ta~ic] d'uo pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|,\\
{}>'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}
d`e \kai%
\eix{\gr{ka`i}} t`hn b'asin t~h| b'asei >'ishn,\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian t~h| gwn'ia|
{}>'ishn <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  iki \"u\c cgende\\ 
iki kenar\\
iki kenara e\c sit ise\\
her biri birine,\\
ve taban tabana e\c sit ise,\\
a\c c\i\ da a\c c\i ya e\c sit olacak,\\
e\c sit do\u grularca\\
i\c cerilen;\\
\ozqed.
}


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\end{proposition}

%\newpage 

\begin{proposition}{9}%Proposition I.9

\parsen{
The\footnotemark\ given rectilineal angle\\
to cut in two.\footnotemark
}
{
T`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ian e>uj'ugrammon\\
d'iqa teme~in%
\eix{\gr{t'emnw}!\gr{teme~in} aor inf act}.
}
{
Verilmi\c s d\"uzkenar a\c c\i y\i\\
ikiye b\"olmek.
}
\myfntext{\label{note:gen}Here the generic article (see note~\ref{note:the} to Proposition 1 above) is particularly appropriate.  Suppose we take a straight line with a point $A$ on it and draw a circle with center $A$ cutting the line at $B$ and $C$.  Then the straight line $BC$ has been bisected at $A$.  In particular, \emph{a} line has been bisected.  But this does not mean we have solved the problem of the present proposition.  In modern mathematical English, the proposition could indeed be `To bisect a rectilineal angle'; but then `a' must be understood as `an arbitrary' or `a given'.  Of course, Euclid does supply this qualification in any case.}
\myfntext{For `cut in two' we could say `bisect'; but in at least one place, in Proposition 12, \gr{d'iqa teme~in} will be separated.}

\parsen{
Let be\\
the given rectilineal angle\\
\gk{BAG}.
}
{
>'Estw\\
<h doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc\\
<h <up`o BAG.
}
{
Olsun\\
verilmi\c s\ d\"uzkenar a\c c\i\\
\gk{BAG}.
}


\parsen{
Then it is necessary\\
to cut it in two.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
a>ut`hn d'iqa teme~in%
\eix{\gr{t'emnw}!\gr{teme~in} aor inf act}.
}
{
\Deidee\\
onun ikiye b\"ol\"unmesi.
}


\parsen{
Suppose there has been chosen\\
on \gk{AB} at random a point \gk D,\\
and there has been taken from \gk{AG}\\
\gk{AE}, equal to \gk{AD},\\
and \gk{DE} has been joined,\\
and  there has been constructed on \gk{DE}\\
an equilateral triangle, \gk{DEZ},\\
and \gk{AZ} has been joined.
}
{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp}\\
{}>ep`i t~hc AB tuq`on shme~ion t`o D,\\
\kai%
\eix{\gr{ka`i}} >afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}\\
{}>ap`o t~hc AG\\
t~h| AD >'ish <h AE,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h DE,\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp} >ep`i t~hc DE\\
tr'igwnon >is'opleuron t`o DEZ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h AZ;
}
{
al\i nm\i\c s olsun\\
 \gk{AB} \"uzerinde rastgele bir \gk D noktas\i,\\
ve ayr\i lm\i\c s\ olsun\\
\gk{AG} do\u grusundan\\
\gk{AD}'ya e\c sit olan \gk{AE},\\
ve \gk{DE} birle\c stirilmi\c s\ olsun,\\
ve  in\c sa edilmi\c s\ olsun \gk{DE} \"uzerinde\\
bir \gk{DEZ} e\c skenar \"u\c cgeni,\\
ve \gk{AZ} birle\c stirilmi\c s\ olsun.
}

\parsen{
I say that\\
angle \gk{BAG} has been cut in two\\
by the \strgt{} \gk{AZ}.\\
For, because \gk{AD} is equal to \gk{AE},\\
and \gk{AZ} is common,\\
then the two, \gk{DA} and \gk{AZ}\\
to the two, \gk{EA} and \gk{AZ},\\
are equal,\\
either to either,\\
and the base \gk{DZ} to the base \gk{EZ}\\
is equal;\\
therefore angle \gk{DAZ}\\
to angle \gk{EAZ}\\
is equal.
}
{
l'egw, <'oti\\
<h <up`o BAG gwn'ia d'iqa t'etmhtai\\
<up`o t~hc AZ e>uje'iac.\\
{}>Epe`i%
\eix{\gr{>epe'i}} g`ar\\
{}>'ish >est`in <h AD t~h| AE,\\
koin`h d`e <h AZ,\\
d'uo d`h%
\eix{\gr{d'h}} a<i DA, AZ\\
dus`i ta~ic EA, AZ >'isai e>is`in\\
<ekat'era <ekat'era|.\\
\kai%
\eix{\gr{ka`i}} b'asic <h DZ\\
b'asei t~h| EZ >'ish >est'in;\\
gwn'ia >'ara\eix{\gr{>'ara}} <h <up`o DAZ\\
gwn'ia| t~h| <up`o EAZ >'ish >est'in.
}
{
\Legohoti\\
 \gk{BAG} a\c c\i s\i\ ikiye b\"ol\"unm\"u\c s oldu\\
  \gk{AZ} do\u grusu taraf\i ndan.\\
\Gar\ oldu\u gundan\\
\gk{AD} \gk{AE}'a e\c sit,\\
ve \gk{AZ} ortak,\\
\dee\ \gk{DA}, \gk{AZ} ikilisi\\
\gk{EA}, \gk{AZ} ikilisine e\c sittir\\
her biri birine,\\
ve \gk{DZ} taban\i\\
\gk{EZ} taban\i na e\c sittir;\\
\ara\  \gk{DAZ} a\c c\i s\i\\
\gk{EAZ} a\c c\i s\i na e\c sittir.
}

\parsen{
Therefore the given rectilineal angle\\
\gk{BAG}\\
has been cut in two\\
by the \strgt{} \gk{AZ};\\
\myqef
}
{
<H >'ara\eix{\gr{>'ara}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc\\
<h <up`o BAG\\
d'iqa t'etmhtai\\
<up`o t~hc AZ e>uje'iac;\\
<'oper >'edei poi~hsai.
}
{
\Ara\ verilmi\c s d\"uzkenar a\c c\i\\
\gk{BAG}\\
ikiye b\"ol\"unm\"u\c s oldu\\
\gk{AZ} do\u grusunca;\\
\ozqef.
}

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\end{proposition}




\begin{proposition}{10}%Proposition I.10

\parsen{
The given bounded \strgt\\
to cut in two.
}
{
T`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian peperasm'enhn%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
d'iqa teme~in%
\eix{\gr{t'emnw}!\gr{teme~in} aor inf act}.
}
{
Verilmi\c s s\i n\i rl\i\ do\u gruyu\\
ikiye b\"olmek.\\
}

\parsen{
Let be\\
the given bounded straight line \gk{AB}.\\
}
{
>'Estw\\
<h doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia peperasm'enh%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
<h AB;
}
{
Olsun\\
verilmi\c s s\i n\i rl\i\ do\u gru\\
\gk{AB}.\\
}

\parsen{
It is necessary then\\
the bounded straight line \gk{AB} to cut in two.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
t`hn AB e>uje~ian peperasm'enhn%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
d'iqa teme~in%
\eix{\gr{t'emnw}!\gr{teme~in} aor inf act}.
}
{
\Deidee\\
\gk{AB} s\i n\i rl\i\ do\u grusunu\\
ikiye b\"olmek.\\
}

\parsen{
Suppose there has been constructed\\
on it\\
an equilateral triangle, \gk{ABG},\\
and suppose has been cut in two\\
the angle \gk{AGB} by the \strgt{} \gk{GD}.
}
{
Sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp} >ep> a>ut~hc\\
tr'igwnon >is'opleuron t`o ABG,\\
\kai%
\eix{\gr{ka`i}} tetm'hsjw%
\eix{\gr{t'emnw}!\gr{tetm'hsjw} 3rd sg perf imperat mp}\\
<h <up`o AGB gwn'ia d'iqa\\
t~h| GD e>uje'ia|;
}
{
\.In\c sa edilmi\c s\ olsun \"uzerinde\\
\gk{ABG} e\c skenar \"u\c cgeni,\\
ve b\"ol\"unm\"u\c s olsun\\
\gk{AGB} a\c c\i s\i\ ikiye\\
\gk{GD} do\u grusunca.\\
}


\parsen{
I say that\\
the \strgt{} \gk{AB} has been cut in two\\
at the point \gk D.\\
For, because \gk{AG} is equal to \gk{AB},\\
and \gk{GD} is common,\\
the two, \gk{AG} and \gk{GD},\\
to the two, \gk{BG}, \gk{GD},\\
are equal,\\
either to either,\\
and angle \gk{AGD}\\
to angle \gk{BGD}\\
is equal;\\
therefore the base \gk{AD} to the base \gk{BD}\\
is equal.
}
{l'egw, <'oti\\
<h AB e>uje~ia d'iqa t'etmhtai\\
kat`a t`o D shme~ion.\\
{}>Epe`i%
\eix{\gr{>epe'i}} g`ar\\
{}>'ish >est`in <h AG t~h| GB,\\
koin`h d`e <h GD,\\
d'uo d`h%
\eix{\gr{d'h}} a<i AG, GD\\
d'uo ta~ic BG, GD >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o AGD\\
gwn'ia| t~h| <up`o BGD >'ish >est'in;\\
b'asic >'ara\eix{\gr{>'ara}} <h AD\\
b'asei t~h| BD >'ish >est'in.
}
{
\Legohoti\\
\gk{AB} do\u grusu ikiye b\"ol\"unm\"u\c s oldu\\
 \gk D noktas\i nda.\\
\Gar\ oldu\u gundan\\
\gk{AG} \gk{AB} kenar\i na e\c sit,\\
ve \gk{GD} ortak,\\
\dee\ \gk{AG} ve \gk{GD} ikilisi\\
\gk{BG}, \gk{GD} ikilisine e\c sittir,\\
her biri birine,\\
ve \gk{AGD} a\c c\i s\i\\
\gk{BGD} a\c c\i s\i na e\c sittir;\\
\ara\  \gk{AD} taban\i\\
\gk{BD} taban\i na e\c sittir.
}

\parsen{
Therefore the given bounded\qquad \strgt,\\
\gk{AB},\\
has been cut in two at \gk D;\\
\myqef
}
{
<H >'ara\eix{\gr{>'ara}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia peperasm'enh%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp}\\
<h AB\\
d'iqa t'etmhtai kat`a t`o D;\\
<'oper >'edei poi~hsai.
}
{
\Ara\ verilmi\c s\ s\i n\i rl\i\\
\gk{AB},\\
\gk D noktas\i nda ikiye b\"ol\"unm\"u\c s oldu;\\
\ozqef.
}

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\end{proposition}



\begin{proposition}{11}%Proposition I.11

\newparsen{
T~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia|\\
{}>ap`o to~u pr`oc a>ut~h| doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou\\
pr`oc >orj`ac gwn'iac\\
e>uje~ian gramm`hn >agage~in%
\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
Verilmi\c s bir do\u gruya\\
\"uzerinde verilmi\c s bir noktadan\\
dik a\c c\i larda\\
bir do\u gru ilerletmek.}

\newparsen{
>'Estw\\
<h m`en%
\eix{\gr{m'en}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h AB\\
t`o d`e doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme~ion >ep> a>ut~hc t`o G;
}
{
Olsun\\
verilmi\c s\  do\u gru \gk{AB},\\
ve \"uzerinde verilmi\c s nokta \gk G.\\
}

\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
{}>ap`o to~u G shme'iou\\
t~h| AB e>uje'ia|\\
pr`oc >orj`ac gwn'iac\\
e>uje~ian gramm`hn >agage~in%
\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
\Deidee\\
\gk G noktas\i ndan\\
\gk{AB} do\u grusuna\\
dik a\c c\i larda\\
bir do\u gru ilerletmek.\\
}


\newparsen{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp}\\
{}>ep`i t~hc AG\\
tuq`on shme~ion t`o D,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| GD >'ish <h GE,\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf imperat mp}\\
{}>ep`i t~hc DE\\
tr'igwnon >is'opleuron t`o ZDE,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h ZG;
}
{
al\i nm\i\c s olsun\\
\gk{AG}'da\\
rastgele bir \gk D noktas\i\\
ve otursun\\
\gk{GD}'ya e\c sit olan \gk{GE},\\
ve in\c sa edilmi\c s\ olsun\\
\gk{DE} \"uzerinde\\
\gk{ZDE} e\c skenar \"u\c cgeni,\\
ve \gk{ZG} birle\c stirilmi\c s\ olsun.\\
}

\newparsen{
l'egw, <'oti\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| AB\\
{}>ap`o to~u pr`oc a>ut~h| doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou to~u G\\
pr`oc >orj`ac gwn'iac\\ 
e>uje~ia gramm`h >~hktai%
\eix{\gr{>'agw}!\gr{>~hktai} 3rd sg perf ind mp} <h ZG.\\
{}>Epe`i%
\eix{\gr{>epe'i}} g`ar >'ish >est`in <h DG t~h| GE,\\
koin`h d`e <h GZ,\\
d'uo d`h%
\eix{\gr{d'h}} a<i DG, GZ\\
dus`i ta~ic EG, GZ >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} b'asic <h DZ\\
b'asei t~h| ZE >'ish >est'in;\\
gwn'ia >'ara\eix{\gr{>'ara}} <h <up`o DGZ\\
gwn'ia| t~h| <up`o EGZ >'ish >est'in;\\
ka'i e>isin >efex~hc.\\
<'otan d`e e>uje~ia\\
{}>ep> e>uje~ian staje~isa\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
t`ac >efex~hc gwn'iac\\
{}>'isac >all'hlaic poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
{}>orj`h <ekat'era t~wn >'iswn gwni~wn >estin;\\
{}>orj`h >'ara\eix{\gr{>'ara}} >est`in\\
<ekat'era t~wn <up`o DGZ, ZGE.
}
{
\Legohoti\\
verilmi\c s \gk{AB} do\u grusuna\\
\"uzerindeki \gk G noktas\i ndan\lli\\
dik a\c c\i larda\\
bir \gk{ZG} do\u grusu ilerletilmi\c s oldu.\\
\Gar\  \gk{DG}, \gk{GE}'a e\c sit oldu\u gundan,\\
ve \gk{GZ} ortak oldu\u gundan,\\
\dee\ \gk{DG} ve \gk{GZ} ikilisi,\\
\gk{EG} ve \gk{GZ} ikilisine e\c sittir,\\
her biri birine;\\
ve \gk{DZ} taban\i\\
\gk{ZE} taban\i na e\c sittir;\\
\ara\ \gk{DGZ} a\c c\i s\i\\
\gk{EGZ} a\c c\i s\i na e\c sittir;\\
ve biti\c siktir.\\
Ne zaman bir do\u gru,\\
bir do\u gru \"uzerine dikilmi\c s,\\
biti\c sik a\c c\i lar\i\\
birbirine e\c sit yaparsa,\\
e\c cit a\c c\i lar\i n her biri, diktir.\\
\Ara\ diktir\\
\gk{DGZ}, \gk{ZGE} a\c c\i lar\i n\i n her biri.\\
}

\newparsen{
T~h| >'ara\eix{\gr{>'ara}} doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| AB\\
{}>ap`o to~u pr`oc a>ut~h| doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou to~u G\\
pr`oc >orj`ac gwn'iac\\
e>uje~ia gramm`h >~hktai%
\eix{\gr{>'agw}!\gr{>~hktai} 3rd sg perf ind mp} <h GZ;\\
<'oper >'edei poi~hsai.
}
{
\Ara,  verilmi\c s  \gk{AB} do\u grusuna,\\
\"uzerinde verilmi\c s\ \gk G noktas\i nda,\lli\\
dik a\c c\i larda,\\
bir \gk{GZ} do\u grusu ilerletilmi\c s oldu;\\
\ozqef.
}

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    \begin{comment}
      

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    \end{comment}
\end{center}
\end{proposition}



\begin{proposition}{12}%Proposition I.12

\newparsen{
>Ep`i t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian >'apeiron\\
{}>ap`o to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou,\\
<`o m'h >estin >ep> a>ut~hc,\\
k'ajeton e>uje~ian gramm`hn >agage~in\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
Verilmi\c s s\i n\i rlanmam\i\c s\ do\u gruya,\\
verilmi\c s bir noktadan,\\
\"uzerinde olmayan,\\
dikey do\u gru bir \c cizgi ilerletmek.\\
}

\newparsen{
>'Estw\\
<h m`en%
\eix{\gr{m'en}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia >'apeiroc\\
<h AB\\
t`o d`e doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme~ion,\\
<`o m'h >estin >ep> a>ut~hc,\\
t`o G;\\
}
{
Olsun\\
verilmi\c s s\i n\i rlanmam\i\c s\ do\u gru\\
\gk{AB},\\
ve verilmi\c s nokta,\\
\"uzerinde olmayan,\\
\gk G.\\
}

\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
{}>ep`i t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian >'apeiron t`hn AB\\
{}>ap`o to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou to~u G,\\
<`o m'h >estin >ep> a>ut~hc,\\
k'ajeton e>uje~ian gramm`hn >agage~in\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
\Deidee\\
verilmi\c s\ s\i n\i rlanmam\i\c s\ \gk{AB} do\u grusuna\\
verilmi\c s\ \gk G noktas\i ndan,\\
\"uzerinde olmayan,\\
dikey do\u gru bir \c cizgi ilerletmek.\\
}

\newparsen{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp} g`ar\\
{}>ep`i t`a <'etera m'erh t~hc AB e>uje'iac\\
tuq`on shme~ion t`o D,\\
\kai%
\eix{\gr{ka`i}} k'entrw| m`en%
\eix{\gr{m'en}} t~w| G\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| GD\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o EZH,\\
\kai%
\eix{\gr{ka`i}} tetm'hsjw%
\eix{\gr{t'emnw}!\gr{tetm'hsjw} 3rd sg perf imperat mp}
<h EH e>uje~ia d'iqa kat`a t`o J,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp}\\
a<i GH, GJ, GE e>uje~iai;
}
{
\Gar\ almi\c s\ olsun\\
\gk{AB} do\u grusunun di\u ger taraf\i nda\\
rastgele bir \gk D noktas\i,\\
ve \gk G merkezinde,\\
 \gk{GD} uzakl\i \u g\i nda,\\
bir \gk{EZH} dairesi \c cizilmi\c s\ olsun,\\
ve \gk{EH} do\u grusu  \gk J noktas\i nda ikiye b\"ol\"unm\"u\c s olsun,\\
ve birle\c stirilmi\c s\ olsun\\
\gk{GH}, \gk{GJ}, ve \gk{GE} do\u grular\i .\\
}

\parsen{
I say that\\
to the given unbounded \strgt\\
\gk{AB},\\
from the given point \gk G,\\
which is not on it,\\
has been drawn a perpendicular, \gk{GJ}.\\
For, because \gk{HJ} is equal to \gk{JE},\\
and \gk{JG} is common,\\
the two, \gk{HJ} and \gk{JG},\\
to the two, \gk{EJ} and \gk{JG}, are equal,\\
either to either;\\
and the base \gk{GH} to the base \gk{GE}\\
is equal;\\
therefore angle \gk{GJH}\\
to angle \gk{EJG}\\
is equal;\\
and they are adjacent.\\
Whenever a \strgt,\\
standing on a \strgt,\\
the adjacent angles\\
equal to one another make,\\
right\\
either of the equal angles is,\\
and\\
the \strgt{} that has been stood\\
is called perpendicular\\
to that on which it has been stood.
}
{
l'egw, <'oti\\
{}>ep`i t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian >'apeiron\\
t`hn AB\\
{}>ap`o to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou to~u G,\\
<`o m'h >estin >ep> a>ut~hc,\\
k'ajetoc >~hktai%
\eix{\gr{>'agw}!\gr{>~hktai} 3rd sg perf ind mp} <h GJ.\\
{}>Epe`i%
\eix{\gr{>epe'i}} g`ar >'ish >est`in <h HJ t~h| JE,\\
koin`h d`e <h JG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i HJ, JG\\
d'uo ta~ic EJ, JG >'isai e<is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} b'asic <h GH\\
b'asei t~h| GE >estin >'ish;\\
gwn'ia >'ara\eix{\gr{>'ara}} <h <up`o GJH\\
gwn'ia| t~h| <up`o EJG >estin >'ish.\\
ka'i e>isin >efex~hc.\\
<'otan d`e e>uje~ia\\
{}>ep> e>uje~ian staje~isa\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
t`ac >efex~hc gwn'iac\\
{}>'isac >all'hlaic poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
{}>orj`h <ekat'era t~wn >'iswn gwni~wn >estin,\\
\kai%
\eix{\gr{ka`i}} <h >efesthku~ia%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>efesthk'wc} part perf act} 
e>uje~ia\\
k'ajetoc kale~itai%
\eix{\gr{kal'ew}}\\
{}>ef> <`hn >ef'esthken%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>ef'esthke} 3rd sg perf ind act}.
}
{
\Legohoti\\
verilmi\c s s\i n\i rlanmam\i\c s\ \gk{AB} do\u grusuna,\\
verilmi\c s \gk G noktas\i ndan,\\
\"uzerinde olmayan,\\
dikey \gk{GJ} ilerletilmi\c s oldu\\
\Gar\ \gk{HJ}, \gk{JE}'a e\c sit oldu\u gundan,\\
ve \gk{JG} ortak oldu\u gundan,\\
o halde \gk{HJ} ve \gk{JG} ikilisi,\\
\gk{EJ} ve \gk{JG} ikilisine e\c sittir,\\
her biri birine;\\
ve \gk{GH} taban\i\\
\gk{GE} taban\i na e\c sittir;\\
\ara\ \gk{GJH} a\c c\i s\i\\
\gk{EJG} a\c c\i s\i na e\c sittir.\\
Ve biti\c siktir.\\
Ne zaman bir do\u gru,\\
bir do\u gru \"uzerinde dikildi\u ginde,\\
biti\c sik a\c c\i lar\i\\
birbirine e\c sit yaparsa,\\
e\c sit a\c c\i lar\i n her biri diktir,\\
ve dikilmi\c s do\u gruya\\
dikey denir\\
\"uzerine dikildi\u gi [do\u gru]ya.\\
}

\newparsen{
>Ep`i t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} >'ara\eix{\gr{>'ara}} e>uje~ian >'apeiron t`hn AB\\
{}>ap`o to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou to~u G,\\
<`o m'h >estin >ep> a>ut~hc,\\
k'ajetoc ~>hktai <h GJ;\\
<'oper >'edei poi~hsai.
}
{
\Ara,  verilmi\c s s\i n\i rlanmam\i\c s\ \gk{AB} do\u gruya,\\
verilmi\c s \gk G noktas\i ndan,\\
\"uzerinde olmayan,\\
dikey \gk{GJ}, ilerletilmi\c s oldu;\\
\ozqef.
}


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\end{proposition}



\begin{proposition}{13}%Proposition I.13

\newparsen{
>E`an e>uje~ia\\
{}>ep> e>uje~ian staje~isa\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
gwn'iac poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
{}>'htoi d'uo >orj`ac\\
{}>`h dus`in >orja~ic >'isac\\
poi'hsei%
\eix{\gr{poi'ew}!\gr{poi'hsei} 3rd sg fut ind}.
}
{
E\u ger bir do\u gru,\\
bir do\u grunun \"uzerine dikilmi\c s,\\
a\c c\i lar yaparsa,\\
ya iki dik\\
ya da iki \dike\ e\c sit\\
{}[onlar\i] yapacak.\\
}

\newparsen{
E>uje~ia g'ar\eix{\gr{g'ar}} tic <h AB\\
{}>ep> e>uje~ian t`hn GD staje~isa%
\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
gwn'iac poie'itw%
\eix{\gr{poi'ew}!\gr{poie'itw} 3rd sg pres imperat act} t`ac <up`o GBA, ABD; 
}
{
\Gar\ bir  \gk{AB} do\u grusu,\\
\gk{GD} do\u grusunun \"uzerine dikilmi\c s,\\
\gk{GBA} ve \gk{ABD} a\c c\i lar\i n\i\ olu\c stursun.}

\newparsen{
l`egw, <'oti\\
a<i <up`o GBA, ABD gwn'iai\\
{}>'htoi d'uo >orja'i e>isin\\
{}>`h dus`in >orja~ic >'isai.
}
{
\Legohoti\\
\gk{GBA} ve \gk{ABD} a\c c\i lar\i\\
ya iki dik a\c c\i d\i r\\
ya da iki dik a\c c\i ya e\c sittir.\\
}

\newparsen{
E>i m`en%
\eix{\gr{m'en}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h <up`o GBA t~h| <up`o ABD,\\
d'uo >orja'i e>isin.
}
{
\Oun\ e\u ger e\c sitse\\
\gk{GBA}, \gk{ABD}'ya,\\
iki dik a\c c\i d\i r.\\
}


\newparsen{
e>i d`e o>'u,\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp}\\
{}>ap`o to~u B shme'iou\\
t~h| GD [e>uje'ia|]\\
pr`oc >orj`ac\\
<h BE;
}
{
E\u ger de\u gilse,\\
ilerletilmi\c s olsun,\\
\gk B noktas\i ndan,\\
\gk{GD} do\u grusuna,\\
dik [a\c c\i]larda,\\
\gk{BE}.\\
}

\parsen{
Therefore \gk{GBE} and \gk{EBD}\\
are two \rgt s;\\
and since \gk{GBE}\\
to the two, \gk{GBA} and \gk{ABE}, is equal\\
let there be added in common \gk{EBD}.\\
Therefore \gk{GBE} and \gk{EBD}\\
to the three, \gk{GBA}, \gk{ABE}, and \gk{EBD},\\
are equal.\\
Moreover,\\
since \gk{DBA}\\
to the two, \gk{DBE} and \gk{EBA}, is equal\\
let there be added in common \gk{ABG};\\
therefore \gk{DBA} and \gk{ABG}\\
to the three, \gk{DBE}, \gk{EBA}, and \gk{ABG},\\
are equal.\\
And \gk{GBE} and \gk{EBD} were shown\\
equal to the same three.\\
And equals to the same\\
are also equal to one another;\\
also, therefore, \gk{GBE} and \gk{EBD}\\
to \gk{DBA} and \gk{ABG} are equal;\\
but \gk{GBE} and \gk{EBD}\\
are two \rgt s;\\
and therefore \gk{DBA} and \gk{ABG}\\
are equal to two \rgt s.
}
{
a<i >'ara\eix{\gr{>'ara}} <up`o GBE, EBD\\
d'uo >orja'i e>isin;\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} <h <up`o GBE\\
dus`i ta~ic <up`o GBA, ABE\\
{}>'ish >est'in,\\
koin`h\\
proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o EBD;\\
a<i >'ara\eix{\gr{>'ara}} <up`o GBE, EBD\\
tris`i ta~ic <up`o GBA, ABE, EBD\\
{}>'isai e>is'in.\\
p'alin\eix{\gr{p'alin}},\\
{}>epe`i%
\eix{\gr{>epe'i}} <h <up`o DBA\\
dus`i ta~ic <up`o DBE, EBA\\
{}>'ish >est'in,\\
koin`h\\
proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o ABG;\\
a<i >'ara\eix{\gr{>'ara}} <up`o DBA, ABG\\
tris`i ta~ic <up`o DBE, EBA, ABG\\
{}>'isai e>is'in.\\
{}>ede'iqjhsan\eix{\gr{de'iknumi}!\gr{>ede'iqjhsan} 3rd pl aor pass} d`e \kai%
\eix{\gr{ka`i}}\\
a<i <up`o GBE, EBD\\
tris`i ta~ic a>uta~ic >'isai;\\
t`a d`e t~w| a>ut~w| >'isa\\
\kai%
\eix{\gr{ka`i}} >all'hloic >est`in >'isa;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o GBE, EBD >'ara\eix{\gr{>'ara}}\\
ta~ic <up`o DBA, ABG >'isai e>is'in;\\
{}>all`a%
\eix{\gr{>all'a}} a<i <up`o GBE, EBD\\
d'uo >orja'i e>isin;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o DBA, ABG >'ara\eix{\gr{>'ara}}\\
dus`in >orja~ic >'isai e>is'in.
}
{
\Ara\ \gk{GBE} ve \gk{EBD},\\
iki diktir;\\
ve \gk{GBE},\\
\gk{GBA} ve \gk{ABE} ikilisine\\
e\c sit  oldu\u gundan,\\
ortak olarak\\
\gk{EBD}, eklensin.\\
\Ara\ \gk{GBE} ve \gk{EBD},\\
\gk{GBA}, \gk{ABE} ve \gk{EBD} \"u\c cl\"us\"une\\
e\c sittir.\\
\Palin\\
\gk{DBA},\\
\gk{DBE} ve \gk{EBA} ikilisine\\
e\c sit oldu\u gundan,\\
ortak olarak\\
\gk{ABG}, eklensin;\\
\ara\ \gk{DBA} ve \gk{ABG},\\
\gk{DBE}, \gk{EBA} ve \gk{ABG} \"u\c cl\"us\"une\\
e\c sittir.\\
Ve ayr\i ca g\"osterilmi\c sti\\
\gk{GBE} ve \gk{EBD}'n\i n\\
ayn\i\ \"u\c cl\"uye e\c sitli\u gi.\\
Ve ayn\i\ \c seye e\c sitler\\
birbirine de e\c sittir;\\
ve, \ara,  \gk{GBE} ve \gk{EBD},\\
\gk{DBA} ve \gk{ABG}'ya e\c sittir;\\
ama \gk{GBE} ve \gk{EBD},\\
iki diktir;\\
ve \ara\ \gk{DBA} ve \gk{ABG}\\
iki \dike\ e\c sittir.\\
}

\parsen{
If, therefore, a \strgt,\\
stood on a \strgt,\\
make angles,\\
either two \rgt s\\
or equal to two \rgt s\\
it will make;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} e>uje~ia\\
{}>ep> e>uje~ian staje~isa\eix{\gr{<'isthmi}!\gr{staje'ic} part aor pass}\\
gwn'iac poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\ 
{}>'htoi d'uo >orj`ac\\
{}>`h dus`in >orja~ic >'isac\\
poi'hsei%
\eix{\gr{poi'ew}!\gr{poi'hsei} 3rd sg fut ind};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  bir do\u gru,\\
bir do\u grunun \"uzerine dikilmi\c s,\\
a\c c\i lar yaparsa,\\
ya iki dik\\
ya da iki \dike\ e\c sit\\
{}[onlar\i] yapacak;\\
\ozqed.
}
\begin{center}
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{
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\usefont{T1}{ptm}{m}{n}
\rput(3.3603125,0.7584375){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(2.0809374,-1.1015625){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(4.8548436,-1.1415625){\gk G}
\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(0.11375,-1.1215625){\gk D}
\end{pspicture} 
}
\end{center}
\end{proposition}



\begin{proposition}{14}%Proposition I.14

\parsen{
If to some \strgt,\\
and at the same point,\\
two \strgt s,\\
not lying to the same parts,\\
the adjacent angles\\
to two \rgt s\\
make equal,\\
on a \strgt\\
will be with one another\\
the \strgt s.
}
{
>E`an pr'oc tini e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw|\\
d'uo e>uje~iai\\
m`h >ep`i t`a a>ut`a m'erh ke'imenai\\
t`ac >efex~hc gwn'iac\\
dus`in >orja~ic >'isac\\
poi~wsin%
\eix{\gr{poi'ew}!\gr{poi~wsin} 3rd pl pres subj},\\
{}>ep> e>uje'iac\\
{}>'esontai >all'hlaic\\
a<i e>uje~iai.
}
{
E\u ger bir do\u gruya,\\
ve ayn\i\ noktas\i nda,\\
iki do\u gru,\\
ayn\i\ taraf\i nda uzanmayan,\\
biti\c sik a\c c\i lar\i\\
iki \dike\ e\c sit\\
yaparsa,\\
bir do\u gruda\\
birbiriyle olacak\\
do\u grular.
}

\parsen{
For, to some \strgt, \gk{AB},\\
and at the same point, \gk B,\\
two \strgt s \gk{BG} and \gk{BD},\\
not lying to the same parts,\\
the adjacent angles\\
\gk{ABG} and \gk{ABD}\\
equal to two \rgt s\\
---suppose they make.
}
{
Pr`oc g'ar\eix{\gr{g'ar}} tini e>uje'ia| t~h| AB\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| B\\
d'uo e>uje~iai a<i BG, BD\\
m`h >ep`i t`a a>ut`a m'erh ke'imenai\\
t`ac >efex~hc gwn'iac t`ac <up`o ABG, ABD\\
d'uo >orja~ic >'isac\\
poie'itwsan%
\eix{\gr{poi'ew}!\gr{poie'itwsan} 3rd pl pres subj};
}
{\Gar\ bir \gk{AB} do\u grusuna,\\
ve \gk B noktas\i nda,\\
iki \gk{BG} ve \gk{BD} do\u grular\i,\\
ayn\i\ taraf\i nda uzanmayan,\\
biti\c sik \gk{ABG} ve \gk{ABD} a\c c\i lar\i\\
iki \dike\ e\c sit\\
yaps\i n.
}

\parsen{
I say that\\
on a \strgt\\
with \gk{GB} is \gk{BD}. 
}
{
l'egw, <'oti\\
{}>ep> e>uje'iac >est`i\\
t~h| GB <h BD.
}
{\Legohoti\\
bir do\u grudad\i r\\
\gk{GB} ile \gk{BD}. 
}

\parsen{
For, if it is not\\
with \gk{BG} on a \strgt,\\
{}[namely] \gk{BD},\\
let there be,\\
with \gk{BG} in a \strgt,\\
\gk{BE}.
}
{
E>i g`ar m'h >esti\\
t~h| BG >ep> e>uje'iac\\
<h BD,\\
{}>'estw\\
t~h| GB >ep> e>uje'iac\\
<h BE.
}
{
\Gar\ e\u ger de\u gilse\\
\gk{BG} ile bir do\u gruda\\
\gk{BD},\\
olsun\\
\gk{BG} ile bir do\u gruda\\
\gk{BE}.
}

\parsen{
For, since the \strgt{} \gk{AB}\\
has stood\footnotemark\ to the \strgt{} \gk{GBE},\\
therefore angles \gk{ABG} and \gk{ABE}\\
are equal to two \rgt s.\\
Also \gk{ABG} and \gk{ABD}\\
are equal to two \rgt s.\\
Therefore \gk{GBA} and \gk{ABE}\\
are equal to \gk{GBA} and \gk{ABD}.\\
In common\\
suppose there has been taken away\\
\gk{GBA};\\
therefore the remainder \gk{ABE}\\
to the remainder \gk{ABD} is equal,\\
the less to the greater;\\
which is impossible.\\
Therefore it is not [the case that]\\
\gk{BE} is on a \strgt{} with \gk{GB}.\\
Similarly we\footnotemark\ shall show that\\
no other [is so], except \gk{BD}.\\
Therefore on a \strgt\\
is \gk{GB} with \gk{BD}.
}
{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} e>uje~ia <h AB\\
{}>ep> e>uje~ian t`hn GBE >ef'esthken%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>ef'esthke} 3rd sg perf ind act},\lli\\
a<i >'ara\eix{\gr{>'ara}} <up`o ABG, ABE gwn'iai\\
d'uo >orja~ic >'isai e>is'in;\\
e>is`i d`e \kai%
\eix{\gr{ka`i}} a<i <up`o ABG, ABD\\
d'uo >orja~ic >'isai;\\
a<i >'ara\eix{\gr{>'ara}} <up`o GBA, ABE\\
ta~ic <up`o GBA, ABD >'isai e>is'in.\\
koin`h\\
{}>afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}
<h <up`o GBA;\\
loip`h >'ara\eix{\gr{>'ara}} <h <up`o ABE\\
loip~h| t~h| <up`o ABD >estin >'ish,\\
<h >el'asswn t~h| me'izoni;\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}}.\\
o>uk >'ara\eix{\gr{>'ara}}\\
{}>ep> e>uje'iac >est`in <h BE t~h| GB.\\
<omo'iwc d`h%
\eix{\gr{d'h}} de'ixomen%
\eix{\gr{de'iknumi}!\gr{de'ixomen} 1st pl fut ind},\\
<'oti\\
o>ud`e >'allh tic pl`hn t~hc BD;\\
{}>ep> e>uje'iac >'ara\eix{\gr{>'ara}} >est`in\\
<h GB t~h| BD.
}
{
\Oun\ \gk{AB} do\u grusu\\
\gk{GBE} do\u grusunun \"uzerine konuldu\u gundan,\\
\ara\  \gk{ABG} ve \gk{ABE} a\c c\i lar\i\\
iki \dike\ e\c sittir.\\
\gk{ABG} ve \gk{ABD} da\\
iki \dike\ e\c sittir.\\
\Ara\ \gk{GBA} ve \gk{ABE},\\
\gk{GBA} ve \gk{ABD}'ya e\c sittir.\\
Ortak olarak\\
\gk{GBA} \c c\i kart\i lmi\c s olsun.\\
\Ara\ \gk{ABE} kalan\i\\
\gk{ABD} kalan\i na e\c sittir,\\
k\"u\c c\"uk olan b\"uy\"u\u ge;\\
\imkansiz.\\
\Ara\  de\u gildir\\
bir do\u gruda \gk{BE}, \gk{GB} ile.\\
Benzer \c sekilde \dee\ g\"osterece\u giz\\
ki\\
hi\c cbiri [\"oyle de\u gildir], \gk{BD} d\i\c s\i nda.\\
\Ara\ bir do\u grudad\i r\\
\gk{GB}, \gk{BD} ile.
}
\myfntext{This seems to be the first use of the first person \emph{plural.}}

\parsen{
If, therefore, to some \strgt,\\
and at the same point,\\
two \strgt s,\\
not lying in the same parts,\\
adjacent angles\\
two right angles\\
make,\\
on a \strgt\\
will be with one another\\
the \strgt s;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} pr'oc tini e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw|\\
d'uo e>uje~iai\\
m`h >ep`i a>ut`a m'erh ke'imenai\\
t`ac >efex~hc gwn'iac\\
dus`in >orja~ic >'isac\\
poi~wsin%
\eix{\gr{poi'ew}!\gr{poi~wsin} 3rd pl pres subj},\\
{}>ep> e>uje'iac\\
{}>'esontai >all'hlaic\\
a<i e>uje~iai;\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  bir do\u gruya,\\
ve ayn\i\ noktas\i nda,\\
iki do\u gru,\\
ayn\i\ taraf\i nda uzanmayan,\\
biti\c sik a\c c\i lar\i\\
iki \dike\ e\c sit\\
yaparsa,\\
bir do\u gruda\\
birbiriyle olacak\\
do\u grular;\\
\ozqed.
}

\begin{center}
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{
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}
\end{center}

\end{proposition}



\begin{proposition}{15}%Proposition I.15

\parsen{
If two \strgt s cut one another,\\
the vertical\footnotemark\ angles\\
they make equal to one another.
}
{
>E`an d'uo e>uje~iai t'emnwsin%
\eix{\gr{t'emnw}!\gr{t'emnwsin} 3rd pl pres subj act} >all'hlac,\\
t`ac kat`a koruf`hn gwn'iac\\
{}>'isac >all'hlaic poio~usin%
\eix{\gr{poi'ew}!\gr{poio~usin} 3rd pl pres ind act}.
}
{
E\u ger iki do\u gru birbirini keserse,\\
ters a\c c\i lar\i\footnotemark\\
birbirine e\c sit yapar.
}
\footnotetext{Yunancada \emph{ba\c staki a\c c\i lar.}}
\myfntext{The Greek is \gr{kat`a koruf`hn}, which might be translated
  as `at a head', just as, in the conclusion of I.10, \gk{AB} has been
  cut in two `at \gk D', \gr{kat`a t`o D}.  But \gr{koruf'h} and the
  Latin \emph{vertex} can both mean \emph{crown of the head,} and in
  anatomical use, the English \emph{vertical} refers to this crown.
  Apollonius uses \gr{koruf'h} for the vertex of a
  cone~\cite[pp.~286--7]{MR13:419b}.} 

\parsen{
For, let the \strgt s \gk{AB} and \gk{GD}\\
cut one another\\
at the point \gk E.
}
{
D'uo g`ar e>uje~iai a<i AB, GD\\
temn'etwsan%
\eix{\gr{t'emnw}!\gr{t'emn'etwsan} 3rd pl pres imperat act} >all'hlac\\
kat`a t`o E shme~ion;
}
{
\Gar\ \gk{AB} ve \gk{GD} do\u grular\i\ \\
birbirini kessin\\
\gk E noktas\i nda.
}

\parsen{
I say that\\
equal are\\
angle \gk{AEG} to \gk{DEB},\\
and \gk{GEB} to \gk{AED}.
}
{
l'egw, <'oti\\
{}>'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} <up`o AEG gwn'ia t~h| <up`o DEB,\\
<h d`e <up`o GEB t~h| <up`o AED.
}
{
\Legohoti\\
e\c sittir\\
\gk{AEG}, \gk{DEB}'ya,\\
ve \gk{GEB}, \gk{AED}'ya.
}

\parsen{
For, since the \strgt{} \gk{AE}\\
has stood to the \strgt{} \gk{GD},\\
making angles \gk{GEA} and \gk{AED},\\
therefore angles \gk{GEA} and \gk{AED}\\
are equal to two \rgt s.\\
Moreover,\\
since the \strgt{} \gk{DE}\\
has stood to the \strgt{} \gk{AB},\\
making angles \gk{AED} and \gk{DEB},\\
therefore angles \gk{AED} and \gk{DEB}\\
are equal to two \rgt s.\\
And \gk{GEA} and \gk{AED} were shown\\
equal to two \rgt s;\\
therefore \gk{GEA} and \gk{AED}\\
are equal to \gk{AED} and \gk{DEB}.\\
In common\\
suppose there has been taken away\\
\gk{AED};\\
therefore the remainder \gk{GEA}\\
is equal to the remainder \gk{BED};\\
similarly it will be shown that\\
also \gk{GEB} and \gk{DEA} are equal.\footnotemark}
{
>Epe`i%
\eix{\gr{>epe'i}} g`ar e>uje~ia <h AE\\
{}>ep> e>uje~ian t`hn GD\\
{}>ef'esthke%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>ef'esthke} 3rd sg perf ind act}\\
gwn'iac poio~usa%
\eix{\gr{poi'ew}!\gr{poi~wn} part pres act} t`ac <up`o GEA, AED,\\
a<i >'ara\eix{\gr{>'ara}} <up`o GEA, AED gwn'iai\\
dus`in >orja~ic >'isai e>is'in.\\
p'alin\eix{\gr{p'alin}},\\
{}>epe`i%
\eix{\gr{>epe'i}} e>uje~ia <h DE\\
{}>ep> e>uje~ian t`hn AB\\
{}>ef'esthke%
\eix{\gr{<'isthmi}!\gr{>ef'isthmi}!\gr{>ef'esthke} 3rd sg perf ind act}\\
gwn'iac poio~usa%
\eix{\gr{poi'ew}!\gr{poi~wn} part pres act} t`ac <up`o AED, DEB,\\
a<i >'ara\eix{\gr{>'ara}} <up`o AED, DEB gwn'iai\\
dus`in >orja~ic >'isai e>is'in.\\
{}>ede'iqjhsan\eix{\gr{de'iknumi}!\gr{>ede'iqjhsan} 3rd pl aor pass} d`e \kai%
\eix{\gr{ka`i}}\\
a<i <up`o GEA, AED\\
dus`in >orja~ic >'isai;\\
a<i >'ara\eix{\gr{>'ara}} <up`o GEA, AED\\
ta~ic <up`o AED, DEB >'isai e>is'in.\\
koin`h\\
{}>afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}\\
<h <up`o AED;\\
loip`h >'ara\eix{\gr{>'ara}} <h <up`o GEA\\
loip~h| t~h| <up`o BED >'ish >est'in;\\
<omo'iwc d`h%
\eix{\gr{d'h}} deiqj'hsetai%
\eix{\gr{de'iknumi}!\gr{deiqj'hsetai} 3rd sg fut ind mp},\\
<'oti\\
\kai%
\eix{\gr{ka`i}} a<i <up`o GEB, DEA >'isai e>is'in.
}
{
\Gar\ \gk{AE} do\u grusu\\
\gk{GD} do\u grusuna\\
dikilmi\c s oldu\u gundan,\\
\gk{GEA} ve \gk{AED} a\c c\i lar\i n\i\ yapan,\\
\ara\ \gk{GEA} ve \gk{AED} a\c c\i lar\i\\
iki \dike\ e\c sittir.\\
\Palin,\\
 \gk{DE} do\u grusu\\
\gk{AB} do\u grusuna\\
dikilmi\c s oldu\u gundan,\\
\gk{AED} ve \gk{DEB} a\c c\i lar\i n\i\ yapan,\\
\ara\ \gk{AED} ve \gk{DEB} a\c c\i lar\i\\
iki \dike\ e\c sittir.\\
Ve g\"osterilmi\c sti\\
\gk{GEA} ve \gk{AED} a\c c\i lar\i n\i n\\
iki \dike\ e\c sitli\u gi,\\
\ara\ \gk{GEA} ve \gk{AED},\\
\gk{AED} ve \gk{DEB}'ya e\c sittir.\\
Ortak olarak\\
\c c\i kart\i lm\i\c s\ olsun\\
\gk{AED};\\
\ara\ \gk{GEA} kalan\i,\\
\gk{BED} kalan\i na e\c sittir;\\
benzer \c sekilde \dee\ g\"osterilecek\\
ki\\
\gk{GEB} a\c c\i s\i\ da \gk{DEA} a\c c\i s\i na e\c sittir.}
\myfntext{This is a rare moment when two things are said to be equal \emph{simply,} and not equal \emph{to one another.}}

\parsen{
If, therefore,\\
two \strgt s cut one another,\\
the vertical angles\\
they make equal to one another;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}}\\
d'uo e>uje~iai t'emnwsin%
\eix{\gr{t'emnw}!\gr{t'emnwsin} 3rd pl pres subj act} >all'hlac,\\
t`ac kat`a koruf`hn gwn'iac\\
{}>'isac >all'hlaic poio~usin%
\eix{\gr{poi'ew}!\gr{poio~usin} 3rd pl pres ind act};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,\\
iki do\u gru birbirini keserse,\\
ters a\c c\i lar\\
birbirine e\c sit yapar\\
\ozqed.
}
\begin{center}
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\end{center}

\end{proposition}



\begin{proposition}{16}%Proposition I.16

\parsen{
One of the sides of any triangle\\
being extended,\\
the exterior angle\\
than either\\
of the interior and opposite angles\\
is greater.
}
{
Pant`oc trig'wnou\\
mi~ac t~wn pleur~wn prosekblhje'ishc%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass}\\
<h >ekt`oc gwn'ia\\
<ekat'erac\\
t~wn >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion gwni~wn\\
me'izwn >est'in.
}
{
Herhangi bir \"u\c cgenin\\
kenarlar\i n\i n biri uzat\i l\i nca,\\
d\i\c s\ a\c c\i,\\
her birinden\\
(i\c c\ ve kar\c s\i t a\c c\i lar\i n)\\
b\"uy\"ukt\"ur.
}

\parsen{
Let there be\\
a triangle, \gk{ABG},\\
and let there have been extended\\
its side \gk{BG}, to \gk D.
}
{
>'Estw\\
tr'igwnon t`o ABG,\\
\kai%
\eix{\gr{ka`i}} prosekbebl'hsjw%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekbebl'hsjw} 3rd sg perf imperat mp}\\
a>uto~u m'ia pleur`a <h BG >ep`i t`o D;
}
{
Olsun\\
\"u\c cgen \gk{ABG},\\
ve uzat\i lm\i\c s\ olsun\\
onun \gk{BG} kenar\i, \gk D noktas\i na.
}

\parsen{
I say that\\
the exterior angle \gk{AGD}\\
is greater\\
than either\\
of the two interior and opposite angles, \gk{GBA} and \gk{BAG}.
}
{
l`egw, <'oti\\
<h >ekt`oc gwn'ia <h <up`o AGD\\
me'izwn >est`in\\
<ekat'erac\\
t~wn >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
t~wn <up`o GBA, BAG gwni~wn.
}
{
\Legohoti\\
 \gk{AGD} d\i\c s\ a\c c\i s\i\\
b\"uy\"ukt\"ur\\
her birinden\\
i\c c\ ve kar\c s\i t\\
\gk{GBA} ve \gk{BAG}  a\c c\i lar\i n\i n.
}

\parsen{
Suppose \gk{AG} has been cut in two at \gk E,\\
and \gk{BE}, being joined,\\
---suppose it has been extended\\
on a \strgt\ to \gk Z,\\
and there has been laid down,\\
equal to \gk{BE},
\gk{EZ},\\
and there has been joined\\
\gk{ZG},\\
and there has been drawn through\\
\gk{AG} to \gk H.
}
{
Tetm'hsjw%
\eix{\gr{t'emnw}!\gr{tetm'hsjw} 3rd sg perf imperat mp} <h AG d'iqa kat`a t`o E,\lli\\
\kai%
\eix{\gr{ka`i}} >epizeuqje~isa <h BE\\
{}>ekbebl'hsjw%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjw} 3rd sg perf imperat mp}
{}>ep> e>uje'iac >ep`i t`o Z,\lli\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}
t~h| BE >'ish\\
<h EZ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw <h ZG,\\
\kai%
\eix{\gr{ka`i}} di'hqjw%
\eix{\gr{>'agw}!\gr{di'agw}!\gr{di'hqjw} 3rd sg perf imperat mp}
<h AG >ep`i t`o H.\lli
}
{
\gk{AG} kenar\i, E noktas\i ndan ikiye b\"ol\"unm\"u\c s olsun,\\
ve, \gk{BE} birle\c stirilince,\\
bir do\u gruda, \gk Z noktas\i na, uzat\i lm\i\c s\ olsun\\
ve \gk{BE} do\u grusuna e\c sit olan otursun\\
\gk{EZ},\\
ve birle\c stirilmi\c s\ olsun \gk{ZG},\\
ve \gk{AG} do\u grusu, \gk H noktas\i na ilerletilmi\c s olsun.\\
}


\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} AE t~h| EG,\\
<h d`e BE t~h| EZ,\\
d'uo d`h%
\eix{\gr{d'h}} a<i AE, EB\\
dus`i ta~ic GE, EZ >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o AEB\\
gwn'ia| t~h| <up`o ZEG >'ish >est'in;\\
kat`a koruf`hn g'ar\eix{\gr{g'ar}};\\
b'asic >'ara\eix{\gr{>'ara}} <h AB\\
b'asei t~h| ZG >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o ABE tr'igwnon\\
t~w| ZEG trig'wnw| >est`in >'ison,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai e>is`in\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
{}>'ish >'ara\eix{\gr{>'ara}} >est`in\\
<h <up`o BAE t~h| <up`o EGZ.\\
me'izwn d'e >estin\\
<h <up`o EGD t~hc <up`o EGZ;\\
me'izwn >'ara\eix{\gr{>'ara}}\\
<h <up`o AGD t~hc <up`o BAE.\\
<Omo'iwc d`h%
\eix{\gr{d'h}}\\
t~hc BG tetmhm'enhc d'iqa\\
deiqj'hsetai%
\eix{\gr{de'iknumi}!\gr{deiqj'hsetai} 3rd sg fut ind mp} \kai%
\eix{\gr{ka`i}} <h <up`o BGH,\\
tout'estin <h <up`o AGD,\\
me'izwn \kai%
\eix{\gr{ka`i}} t~hc <up`o ABG.
}
{
\Oun\ e\c sit oldu\u gundan\\
\gk{AE}, \gk{EG} do\u grusuna,\\
ve \gk{BE},  \gk{EZ} do\u grusuna,\\
\dee\ \gk{AE} ve \gk{EB} ikilisi,\\
\gk{GE} ve \gk{EZ} ikilisine e\c sittir,\\
her biri birine;\\
ve \gk{AEB} a\c c\i s\i,\\
\gk{ZEG} a\c c\i s\i na e\c sittir,\\
\gar\ ters;\\
\ara\ \gk{AB} taban\i\\
\gk{ZG} taban\i na e\c sittir,\\
ve \gk{ABE} \"u\c cgeni\\
\gk{ZEG} \"u\c cgenine e\c sittir,\\
ve \kalan\ a\c c\i lar\\
\kalan\ a\c c\i lar\i na e\c sittir,\\
her biri birine,\\
(yani) e\c sit kenarlar\i\ g\"orenler.\\
\Ara\ e\c sittir\\
\gk{EGD} ve \gk{EGZ}.\\
Ama b\"uy\"ukt\"ur\\
\gk{BAE}, \gk{EGZ} a\c c\i s\i ndan;\\
\ara\ b\"uy\"ukt\"ur\\
\gk{AGD}, \gk{BAE} a\c c\i s\i ndan.\\
Benzer \c sekilde \dee\\
ikiye b\"ol\"unm\"u\c s oldu\u gundan \gk{BG},\\
g\"osterilecek ki \gk{BGH},\\
\gk{AGD} a\c c\i s\i na e\c sit olan,\\
b\"uy\"ukt\"ur \gk{ABG} a\c c\i s\i ndan da.
}

\parsen{
Therefore, of any triangle,\\
one of the sides\\
being extended,\\
the exterior angle\\
than either\\
of the interior and opposite angles\\
is greater;\\
\myqed
}
{
Pant`oc >'ara\eix{\gr{>'ara}} trig'wnou\\
mi~ac t~wn pleur~wn\\
prosekblhje'ishc%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass}\\
<h >ekt`oc gwn'ia \\
ekat'erac\\
t~wn >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion gwni~wn\\
me'izwn >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir \"u\c cgenin,\\
kenarlar\i ndan biri\\
uzat\i ld\i \u g\i nda,\\
d\i\c s\ a\c c\i\\
her bir\\
i\c c\ ve kar\c s\i t a\c c\i dan\\
b\"uy\"ukt\"ur;\\
\ozqed.
}

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\end{proposition}


\begin{proposition}{17}%Proposition I.17

\parsen{
Two angles of any triangle\\
are less than two \rgt s\\
---taken anyhow.
}
{
Pant`ovc trig'wnou a<i d'uo gwn'iai\\
d'uo >orj~wn >el'asson'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp}.
}
{
Herhangi bir \"u\c cgenin iki a\c c\i s\i\\
k\"u\c c\"ukt\"ur iki dik a\c c\i dan,\\
\nasilalinirsa.
}

\parsen{
Let there be\\
a triangle, \gk{ABG}.
}
{
>'Estw\\
tr'igwnon t`o ABG;
}
{
Olsun\\
\"u\c cgen \gk{ABG}.
}


\parsen{
I say that\\
two angles of triangle \gk{ABG}\\
are less than two \rgt s\\
---taken anyhow.
}
{
>l'egw, <'oti\\
to~u ABG trig'wnou\\
a<i d'uo gwn'iai\\
d'uo >orj~wn >el'atton'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp}.
}
{
\Legohoti\\
\gk{ABG} \"u\c cgeninin\\
iki a\c c\i s\i\\
k\"u\c c\"ukt\"ur iki dik a\c c\i dan,\\
\nasilalinirsa.
}


\parsen{
For, suppose there has been extended\\
\gk{BG} to \gk D.
}
{
>Ekbebl'hsjw%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjw} 3rd sg perf imperat mp} g`ar\\
<h BG >ep`i t`o D.
}
{
\Gar\ uzat\i lm\i\c s\ olsun\\
\gk{BG}, \gk D'ya.
}

\parsen{
And since, of triangle \gk{ABG},\\
\gk{AGD} is an exterior angle,\\
it is greater\\
than the interior and opposite \gk{ABG}.\\
Let \gk{AGB} be added in common;\\
therefore \gk{AGD} and \gk{AGB}\\
are greater than \gk{ABG} and \gk{BGA}.\\
But \gk{AGD} and \gk{AGB}\\
are equal to two \rgt s;\\
therefore \gk{ABG} and \gk{BGA}\\
are less than two \rgt s.\\
Similarly we shall show that\\
also \gk{BAG} and \gk{AGB}\\
are less than two \rgt s,\\
and yet [so are] \gk{GAB} and \gk{ABG}.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} trig'wnou to~u ABG\\
{}>ekt'oc >esti gwn'ia\\
<h <up`o AGD,\\
me'izwn >est`i\\
t~hc >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
t~hc <up`o ABG.\\
koin`h\\
proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o AGB;\\
a<i >'ara\eix{\gr{>'ara}} <up`o AGD, AGB\\
t~wn <up`o ABG, BGA me'izon'ec e>isin.\\
{}>all>%
\eix{\gr{>all'a}} a<i <up`o AGD, AGB\\
d'uo >orja~ic >'isai e>is'in;\\
a<i >'ara\eix{\gr{>'ara}} <up`o ABG, BGA\\
d'uo >orj~wn >el'asson'ec e>isin.\\
<omo'iwc d`h%
\eix{\gr{d'h}} de'ixomen%
\eix{\gr{de'iknumi}!\gr{de'ixomen} 1st pl fut ind act},\\
<'oti\\
\kai%
\eix{\gr{ka`i}} a<i <up`o BAG, AGB\\
d'uo >orj~wn >el'asson'ec e>isi\\
\kai%
\eix{\gr{ka`i}} >'eti%
\eix{\gr{>eti}} 
a<i <up`o GAB, ABG.
}
{
Ve \gk{ABG} \"u\c cgeninin\\
d\i\c s\ a\c c\i s\i\ oldu\u gundan\\
\gk{AGD}\\
b\"uy\"uk\\
i\c c\ ve kar\c s\i t\\
\gk{ABG} a\c c\i s\i ndan.\\
Ortak olarak\\
\gk{AGB}, eklensin;\\
\ara\ \gk{AGD} ve \gk{AGB},\\
\gk{ABG} ve \gk{BGA}'dan b\"uy\"ukt\"ur.\\
Ama \gk{AGD} ve \gk{AGB},\\
iki dik a\c c\i ya e\c sittir;\\
\ara\ \gk{ABG} ve \gk{BGA},\\
iki dik a\c c\i dan k\"u\c c\"ukt\"ur.\\
Benzer \c sekilde \dee\ g\"osterece\u giz\\
ki\\
\gk{BAG} ve \gk{AGB} de\\
iki dik a\c c\i dan k\"u\c c\"ukt\"ur,\\
ve sonra \gk{GAB} ve \gk{ABG} [\"oyledir].
}

\parsen{
Therefore two angles of any triangle\\
are greater than two \rgt s\\
---taken anyhow;\\
\myqed
}
{
Pant`ovc >'ara\eix{\gr{>'ara}} trig'wnou\\
a<i d'uo gwn'iai\\
d'uo >orj~wn >el'ass\-on'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp};\\
<'oper >'edei de~ixai.
}
{
\Ara\ herhangi bir \"u\c cgenin\\
iki a\c c\i s\i\\
iki dik a\c c\i dan k\"u\c c\"ukt\"ur,\\
\nasilalinirsa;\\
\ozqed.
}
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\end{proposition}





\begin{proposition}{18}%Proposition I.18

\parsen{
Of any triangle,\\
the greater side\\
subtends the greater angle.\footnotemark
}
{
Pant`oc trig'wnou\\
<h me'izwn pleur`a\\
t`hn me'izona gwn'ian\\
<upote'inei%
\eix{\gr{te'inw}!\gr{<upote'inw}}.
}
{
Herhangi bir \"u\c cgende\\
daha b\"uy\"uk bir kenar,\\
daha b\"uy\"uk bir a\c c\i y\i\\
rapteder.
}

\parsen{
For, let there be\\
a triangle, \gk{ABG},\\
having side \gk{AG} greater than \gk{AB}.
}
{
>'Estw g`ar\\
tr'igwnon t`o ABG\\
me'izona >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
t`hn AG pleur`an\\
t~hc AB; 
}
{
\Gar\ olsun\\
\"u\c cgen \gk{ABG},\\
daha b\"uy\"uk olan\\
\gk{AG} kenar\i\\
\gk{AB}'dan.
}

\parsen{
I say that\\
also angle \gk{ABG}\\
is greater than \gk{BGA}.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ABG\\
me'izwn >est`i\\
t~hc <up`o BGA;
}
{
\Legohoti\\
\gk{ABG} a\c c\i s\i\ da\\
daha b\"uy\"ukt\"ur\\
\gk{BGA} a\c c\i s\i ndan.\\
}

\parsen{
For, since \gk{AG} is greater than \gk{AB},\\
suppose there has been laid down,\\
equal to \gk{AB},\\
\gk{AD},\\
and let \gk{BD} be joined.
}
{
>Epe`i%
\eix{\gr{>epe'i}} g`ar me'izwn >est`in <h AG t~hc AB,\lli\\
ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| AB >'ish <h AD,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h BD.
}
{
\Gar\ \gk{AG}, \gk{AB} kenar\i ndan daha b\"uy\"uk oldu\u gundan,\\
otursun\\
\gk{AB}'ya e\c sit olan \gk{AD},\\
ve birle\c stirilmi\c s\ olsun \gk{BD}.
}

\parsen{
Since also, of triangle \gk{BGD},\\
angle \gk{ADB} is exterior,\\
it is greater\\
than the interior and opposite \gk{DGB};\\
and \gk{ADB} is equal to \gk{ABD},\\
since side \gk{AB} is equal to \gk{AD};\\
greater therefore\\
is \gk{ABD} than \gk{AGB};\\
by much, therefore,\\
\gk{ABG} is greater\\
than \gk{AGB}.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} trig'wnou to~u BGD\\
{}>ekt'oc >esti gwn'ia <h <up`o ADB,\\
me'izwn >est`i\\
t~hc >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
t~hc <up`o DGB;\\
{}>'ish d`e <h <up`o ADB t~h| <up`o ABD,\\
{}>epe`i%
\eix{\gr{>epe'i}} \kai%
\eix{\gr{ka`i}} pleur`a <h AB\\
t~h| AD >estin >'ish;\\
me'izwn >'ara\eix{\gr{>'ara}}\\
\kai%
\eix{\gr{ka`i}} <h <up`o ABD t~hc <up`o AGB;\\
poll~w| >'ara\eix{\gr{>'ara}} <h <up`o ABG me'izwn >est`i t~hc <up`o AGB.
}
{
\gk{BGD} \"u\c cgeninin\\
d\i\c s\ a\c c\i\ oldu\u gundan \gk{ADB} a\c c\i s\i\ da,\\
b\"uy\"ukt\"ur\\
i\c c\ ve kar\c s\i t\\
\gk{DGB} a\c c\i s\i ndan;\\
ve \gk{ADB}, \gk{ABD}'ya e\c sittir,\\
\gk{AB} kenar\i\ da,\\
\gk{AD}'ya e\c sit oldu\u gundan;\\
\ara\ b\"uy\"ukt\"ur\\
\gk{ABD} da, \gk{AGB}'dan;\\
\ara\ \gk{ABG}, \gk{AGB} a\c c\i s\i ndan \c cok daha b\"uy\"ukt\"ur.
}

\parsen{
Therefore, of any triangle,\\
the greater side\\
subtends the greater angle;\\
\myqed
}
{
Pant`oc >'ara\eix{\gr{>'ara}} trig'wnou\\
<h me'izwn pleur`a\\
t`hn me'izona gwn'ian\\
<upote'inei%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir \"u\c cgende\\
daha b\"uy\"uk bir kenar,\\
daha b\"uy\"uk bir a\c c\i y\i\\
rapteder;\\
\ozqed.
}

\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.1345313)(5.5321875,1.1345313)
\psline[linewidth=0.04cm](5.2771873,-0.80921876)(1.0571876,-0.80921876)
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(2.2171874,0.37578124){\gk D}
\end{pspicture} 
}

\end{center}

\end{proposition}



\begin{proposition}{19}%Proposition I.19

\parsen{
Of any triangle,\\
under the greater angle\\
the greater side subtends.\footnotemark
}
{
Pant`oc trig'wnou\\
<up`o t`hn me'izona gwn'ian\\
<h me'izwn pleur`a\\
<upote'inei%
\eix{\gr{te'inw}!\gr{<upote'inw}}.
}
{
Herhangi bir \"u\c cgende,\\
daha b\"uy\"uk bir a\c c\i,\\
daha b\"uy\"uk bir kenar taraf\i ndan\\
raptedilir.
}

\myfntext{Heath here uses the expedient of the passive:  `The greater
  angle is subtended by the greater side.'}

\parsen{
For, let there be\\
a triangle, \gk{ABG},\\
having angle \gk{ABG} greater\\
than \gk{BGA}.
}
{
>'Estw g`ar\\
tr'igwnon t`o ABG\\
me'izona >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
t`hn <up`o ABG gwn'ian\\
t~hc <up`o BGA;
}
{
\Gar\ olsun\\
bir \gk{ABG} \"u\c cgeni,\\
daha b\"uy\"uk olan\\
\gk{ABG} a\c c\i s\i\\
\gk{BGA} a\c c\i s\i ndan.
}

\parsen{
I say that\\
also side \gk{AG}\\
is greater than side \gk{AB}.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} pleur`a <h AG\\
pleur~ac t~hc AB\\
me'izwn >est'in.
}
{
\Legohoti\\
\gk{AG} kenar\i\ da\\
\gk{AB} kenar\i ndan\\
daha b\"uy\"ukt\"ur.
}

\parsen{
For if not,\\
either \gk{AG} is equal to \gk{AB}\\
or less;\\
{}[but] \gk{AG} is not equal to \gk{AB};\\
for [if it were],\\
also \gk{ABG} would be\footnotemark\ equal to \gk{AGB};\\
but it is not;\\
therefore \gk{AG} is not equal to \gk{AB}.\\
Nor is \gk{AG} less than \gk{AB};\\
for [if it were],\\
also angle \gk{ABG} would be [less]\\
 than \gk{AGB};\\
but it is not;\\
therefore \gk{AG} is not less than \gk{AB}.\\
And it was shown that\\
it is not equal.\\
Therefore \gk{AG} is greater than \gk{AB}.
}
{
E>i g`ar m'h,\\
{}>'htoi >'ish >est`in\\
<h AG t~h| AB\\
{}>`h >el'asswn;\\
{}>'ish m`en%
\eix{\gr{m'en}} o>~un\eix{\gr{o>~un}} o>uk >'estin\\
<h AG t~h| AB;\\
{}>'ish g`ar >`an >~hn\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ABG\\
t~h| <up`o AGB;\\
o>uk >'esti d'e;\\
o>uk >'ara\eix{\gr{>'ara}} >'ish >est`in\\
<h AG t~h| AB.\\
o>ud`e m`hn%
\eix{\gr{m'hn}} >el'asswn >est`in\\
<h AG t~hc AB;\\
{}>el'asswn g`ar >`an >~hn \kai%
\eix{\gr{ka`i}}\\
gwn'ia <h <up`o ABG\\
t~hc <up`o AGB;\\
o>uk >'esti d'e;\\
o>uk >'ara\eix{\gr{>'ara}} >el'asswn >est`in\\
<h AG t~hc AB.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d'e, <'oti\\
o>ud`e >'ish >est'in.\\
me'izwn >'ara\eix{\gr{>'ara}} >est`in\\
<h AG t~hc AB.
}
{
\Gar\ de\u gil ise,\\
ya e\c sittir\\
\gk{AG}, \gk{AB}'ya\\
ya da daha k\"u\c c\"ukt\"ur.\\
Ama \oun\ e\c sit de\u gildir\\
\gk{AG}, \gk{AB}'ya;\\
\gar \ e\u ger e\c sit olsayd\i,\\
\gk{ABG} a\c c\i s\i\ da,\\
\gk{AGB}'ya [e\c sit olurdu];\\
ama de\u gildir;\\
\ara\ e\c sit de\u gildir\\
\gk{AG}, \gk{AB}'ya.\\
\Meen\ k\"u\c c\"uk de\u gildir\\
\gk{AG}, \gk{AB}'dan;\\
\gar\ e\u ger k\"u\c c\"uk olsayd\i,\\
\gk{ABG} a\c c\i s\i\ da\\
\gk{AGB}'dan [k\"u\c c\"uk olurdu];\\
ama de\u gildir;\\
\ara\ k\"u\c c\"uk de\u gildir\\
\gk{AG}, \gk{AB}'dan.\\
Ve g\"osterilmi\c sti ki\\
e\c sit de\u gildir.\\
\Ara\ daha b\"uy\"ukt\"ur\\
\gk{AG}, \gk{AB}'dan.
}
\myfntext{Literally `was'; but
  this conditional use of \emph{was} is archaic in English.} 

\parsen{
Therefore, of any triangle,\\
under the greater angle\\
the greater side subtends;\\
\myqed
}
{
Pant`oc  >'ara\eix{\gr{>'ara}} trig'wnou\\
<up`o t`hn me'izona gwn'ian\\
<h me'izwn pleur`a\\
<upote'inei;\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir \"u\c cgende,\\
daha b\"uy\"uk bir a\c c\i,\\
daha b\"uy\"uk bir kenar taraf\i ndan\\
raptedilir;\\
\ozqed.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.8145312)(2.17,1.8145312)
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(1.9751563,-1.6642188){\gk G}
\end{pspicture} 
}

\end{center}
\end{proposition}



\begin{proposition}{20}%Proposition I.20

\parsen{
Two sides of any triangle\\
are greater than the remaining one\\
---taken anyhow.
}
{
Pant`oc trig'wnou\\
a<i d'uo pleura`i\\
t~hc loip~hc me'izon'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp}.
}
{
Herhangi bir \"u\c cgenin\\
iki kenar\i\\
\kalan dan daha b\"uy\"ukt\"ur,\\
\nasilalinirsa.
}


\parsen{
For, let there be\\
a triangle, \gk{ABG}.
}
{
>'Estw g`ar\\
tr'igwnon t`o ABG; 
}
{
\Gar\ olsun\\
\"u\c cgen \gk{ABG}.
}

\newparsen{
l'egw, <'oti\\
to~u ABG trig'wnou
a<i d'uo pleura`i\\
t~hc loip~hc me'izon'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp},\\
a<i m`en%
\eix{\gr{m'en}} BA, AG t~hc BG,\\
a<i d`e AB, BG t~hc AG,\\
a<i d`e BG, GA t~hc AB.
}
{
\Legohoti\\
\gk{ABG} \"u\c cgeninin
iki kenar\i\\
\kalan dan daha b\"uy\"ukt\"ur,\\
\nasilalinirsa,\\
\gk{BA} ve \gk{AG}, \gk{BG}'dan,\\
ve \gk{AB} ve \gk{BG}, \gk{AG}'dan,\\
ve \gk{BG} ve \gk{GA}, \gk{AB}'dan.
}

\newparsen{
Di'hqjw%
\eix{\gr{>'agw}!\gr{di'agw}!\gr{di'hqjw} 3rd sg perf imperat mp} g`ar\\
<h BA >ep`i t`o D shme~ion,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}
t~h| GA >'ish <h AD,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw
<h DG.
}
{
\Gar\ ilerletilmi\c s olsun\\
\gk{BA}, \gk D noktas\i na,\\
ve \gk{AD}, \gk{GA}'ya e\c sit otursun,\\
ve \gk{DG} birle\c stirilmi\c s\ olsun.\\
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in <h DA t~h| AG,\lli\\
{}>'ish >est`i \kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ADG\\
t~h| <up`o AGD;\\
me'izwn >'ara\eix{\gr{>'ara}} <h <up`o BGD\\
t~hc <up`o ADG;\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} tr'igwn'on >esti t`o DGB\\
me'izona >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} t`hn <up`o BGD gwn'ian\\
t~hc <up`o BDG,\\  
<up`o d`e t`hn me'izona gwn'ian\\
<h me'izwn pleur`a\\
<upote'inei%
\eix{\gr{te'inw}!\gr{<upote'inw}},\\
<h DB >'ara\eix{\gr{>'ara}} t~hc BG >esti me'izwn.\\
{}>'ish d`e <h DA t~h| AG;\\
me'izonec >'ara\eix{\gr{>'ara}} a<i BA, AG\\
t~hc BG;\\
<omo'iwc d`h%
\eix{\gr{d'h}} de'ixomen%
\eix{\gr{de'iknumi}!\gr{de'ixomen} 1st pl fut ind act}, <'oti\\
\kai%
\eix{\gr{ka`i}} a<i m`en%
\eix{\gr{m'en}} AB, BG t~hc GA\\
me'izon'ec e>isin,\\
a<i d`e BG, GA t~hc AB.
}
{
\Oun\ \gk{DA}, \gk{AG}'ya e\c sit oldu\u gundan,\\
\gk{ADG} de e\c sittir\\
\gk{AGD}'y.\\
\Ara\ \gk{BGD}, b\"uy\"ukt\"ur\\
\gk{ADG}'dan.\\
\gk{DGB} \"u\c cgeninde,\\
\gk{BGD} a\c c\i s\i\ daha b\"uy\"uk oldu\u gundan\\
\gk{BDG}'dan,\\
ve daha b\"uy\"uk a\c c\i,\\
daha b\"uy\"uk kenarca\\
raptedildi\u gindan,\\
\ara\ \gk{DB}, \gk{BG}'dan b\"uy\"ukt\"ur.\\
Ve \gk{DA}, \gk{AG}'ya  e\c sittir;\\
\ara\ \gk{BA} ve \gk{AG} b\"uy\"ukt\"ur\\
\gk{BG}'dan;\\
benzer \c sekilde g\"osterece\u giz ki\\
\gk{AB} ve \gk{BG}, \gk{GA}'dan\\
b\"uy\"ukt\"ur,\\
ve \gk{BG} ve \gk{GA}, \gk{AB}'dan.
}

\newparsen{
Pant`oc >'ara\eix{\gr{>'ara}} trig'wnou\\
a<i d'uo pleura`i\\
t~hc loip~hc me'izon'ec e>isi\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp};\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir \"u\c cgenin\\
iki kenar\i\\
\kalan dan daha b\"uy\"ukt\"ur,\\
\nasilalinirsa;\\
\ozqed.
}

\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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}

\end{center}
\end{proposition}

\begin{proposition}{21}%Proposition I.21

\newparsen{
>E`an trig'wnou\\
{}>ep`i mi~ac t~wn pleur~wn\\
{}>ap`o t~wn per'atwn\\
d'uo e>uje~iai\\
{}>ent`oc sustaj~wsin%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaj~wsi} 3rd pl pres subj pass},\\
a<i sustaje~isai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaje'is} part aor pass}\\
t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn\\
{}>el'attonec m`en%
\eix{\gr{m'en}} >'esontai,\\
me'izona d`e gwn'ian peri'exousin%
\eix{\gr{peri'eqw}!\gr{peri'exousi} 3rd pl fut ind act}.
}
{
E\u ger bir \"u\c cgende,\\
kenarlar\i ndan birinin \"uzerinde,\\
s\i n\i rlardan,\\
iki do\u gru\\
i\c ceride in\c sa edilirse,\\
in\c sa edilmi\c s do\u grular,\\
\"u\c cgenin \kalan\\
iki kenar\i ndan\\
daha k\"u\c c\"uk olacak,\\
ama daha b\"uy\"uk bir a\c c\i y\i\ i\c cerecek.
}

\begin{center}
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\rput(1.9760938,0.853125){\gk E}
\end{pspicture} 
\end{center}


\newparsen{
Trig'wnou g`ar to~u ABG\\
{}>ep`i mi~ac t~wn pleur~wn t~hc BG\\
{}>ap`o t~wn per'atwn t~wn B, G\\
d'uo e>uje~iai >ent`oc sunest'atwsan%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atwsan} 3rd pl perf
  imperat mp}
a<i BD, DG;
}
{
\Gar\ \gk{ABG} \"u\c cgeninin,\\
\gk{BG} kenar\i n\i n \"uzerinde\\
\gk B ve \gk G s\i n\i rlar\i ndan,\\
i\c ceride iki \gk{BD} ve \gk{DG} do\u grular\i\ in\c sa edilmi\c s\ olsun.
}

\newparsen{
l'egw, <'oti\\
a<i BD, DG\\
t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn t~wn BA, AG\\
{}>el'assonec m'en e>isin,\\
me'izona d`e gwn'ian peri'eqousi%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act}\\
t`hn <up`o BDG t~hc <up`o BAG.
}
{
\Legohoti\\
\gk{BD} ve \gk{DG}\\
\"u\c cgenin \kalan\ iki\\
\gk{BA} ve \gk{AG} kenar\i ndan,\\
daha k\"u\c c\"ukt\"ur,\\
ama daha b\"uy\"uk a\c c\i y\i\ i\c cerir:\\
\gk{BDG}, \gk{BAG}'dan [daha b\"uy\"ukt\"ur].
}


\newparsen{
Di'hqjw%
\eix{\gr{>'agw}!\gr{di'agw}!\gr{di'hqjw} 3rd sg perf imperat mp} g`ar <h BD\\
>ep`i t`o E.
}
{
\Gar\ \gk{BD}, ilerletilmi\c s olsun\\
\gk E'a do\u gru.\\
}


\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} pant`oc trig'wnou\\
a<i d'uo pleura`i t~hc loip~hc\\
me'izon'ec e>isin,\\
to~u ABE >'ara\eix{\gr{>'ara}} trig'wnou\\
a<i d'uo pleura`i a<i AB, AE\\
t~hc BE me'izon'ec e>isin;\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp}
<h EG;\\
a<i >'ara\eix{\gr{>'ara}} BA, AG\\
t~wn BE, EG me'izon'ec e>isin.\\
p'alin\eix{\gr{p'alin}},
{}>epe`i%
\eix{\gr{>epe'i}} to~u GED trig'wnou\\
a<i d'uo pleura`i a<i GE, ED\\
t~hc GD me'izon'ec e>isin,\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp}
<h DB;\\
a<i GE, EB >'ara\eix{\gr{>'ara}}\\
t~wn GD, DB me'izon'ec e>isin.\\
{}>all`a%
\eix{\gr{>all'a}} t~wn BE, EG\\
me'izonec >ede'iqjhsan%
\eix{\gr{de'iknumi}!\gr{>ede'iqjhsan} 3rd pl aor pass}\\
a<i BA, AG;\\
poll~w| >'ara\eix{\gr{>'ara}} a<i BA, AG
t~wn BD, DG me'izon'ec e>isin.
}
{
Ve herhangi bir \"u\c cgenin\\
iki kenar\i, \kalan dan\\
b\"uy\"uk oldu\u gundan,\\
\gk{ABE} \"u\c cgeninin,\\
iki \gk{AB} ve \gk{AE} kenarlar\i,\\
\gk{BE} kenar\i ndan b\"uy\"ukt\"ur;\\
ortak olarak \gk{EG} eklensin;\\
\ara\ \gk{BA} ve \gk{AG},\\
\gk{BE} ve \gk{EG}'dan b\"uy\"ukt\"ur.\\
\Palin, \gk{GED} \"u\c cgeninin,\\
iki \gk{GE} ve \gk{ED} kenarlar\i,\\
\gk{GD}'dan b\"uy\"uk oldu\u gundan,\\
ortak olarak \gk{DB} eklenmi\c s\ olsun;\\
\ara\ \gk{GE} ve \gk{EB},\\
\gk{GD} ve \gk{DB}'dan b\"uy\"ukt\"ur.\\
Ama \gk{BE} ve \gk{EG}'dan\\
daha b\"uy\"uk g\"osterilmi\c sti\\
\gk{BA} ve \gk{AG};\\
\ara\ \gk{BA} ve \gk{AG},
 \gk{BD} ve \gk{DG}'dan \c cok daha b\"uy\"ukt\"ur.\\
}


\newparsen{
P'alin,\\
{}>epe`i%
\eix{\gr{>epe'i}} pant`oc trig'wnou
<h >ekt`oc gwn'ia\\
t~hc >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
me'izwn >est'in,\\
to~u GDE >'ara\eix{\gr{>'ara}} trig'wnou\\
<h >ekt`oc gwn'ia <h <up`o BDG\\
me'izwn >est`i t~hc <up`o GED.\\
di`a\eix{\gr{di'a}} ta>ut`a to'inun\\
\kai%
\eix{\gr{ka`i}} to~u ABE trig'wnou\\
<h >ekt`oc gwn'ia <h <up`o GEB\\
me'izwn >est`i t~hc <up`o BAG.\\
{}>all`a%
\eix{\gr{>all'a}} t~hc <up`o GEB\\
me'izwn >ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass}\\
<h <up`o BDG;\\
poll~w| >'ara\eix{\gr{>'ara}}
<h <up`o BDG me'izwn >est`i t~hc <up`o BAG.
}
{
\Palin,\\
herhangi bir \"u\c cgenin
d\i\c s\ a\c c\i s\i\\
i\c c\ ve kar\c s\i t a\c c\i s\i ndan\\
daha b\"uy\"ukt\"ur,\\
\ara,   \gk{GDE} \"u\c cgeninin\\
d\i\c s\ a\c c\i s\i\ \gk{BDG}\\
\gk{GED}'dan b\"uy\"ukt\"ur.\\
\Diatauta\ \toinun,\\
\gk{ABE} \"u\c cgeninin\\
\gk{GEB} d\i\c s\ a\c c\i s\i\ da\\
\gk{BAG}'dan b\"uy\"ukt\"ur.\\
Ama \gk{GEB}'dan,\\
daha b\"uy\"uk g\"osterilmi\c sti\\
\gk{BDG};\\
\ara\ \gk{BDG}, \gk{BAG}'dan \c cok daha b\"uy\"ukt\"ur.
}

\newparsen{
>E`an >'ara\eix{\gr{>'ara}} trig'wnou\\
{}>ep`i mi~ac t~wn pleur~wn\\
{}>ap`o t~wn per'atwn\\
d'uo e>uje~iai\\
{}>ent`oc sustaj~wsin%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaj~wsi} 3rd pl pres subj pass},\\
a<i sustaje~isai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sustaje'is} part aor pass}\\
t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn\\
{}>el'attonec m'en e>isin,\\
me'izona d`e gwn'ian peri'eqousin%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  bir \"u\c cgenin,\\
kenarlar\i ndan birinin\\
s\i n\i rlar\i ndan,\\
iki do\u gru\\
i\c ceride in\c sa edilirse,\\
in\c sa edilen do\u grular,\\
\"u\c cgenin \kalan\\
iki kenar\i ndan\\
daha k\"u\c c\"ukt\"ur,\\
ama daha b\"uy\"uk bir a\c c\i y\i\ i\c cerir;\\
\ozqed.
}

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\rput(1.9760938,0.853125){\gk E}
\end{pspicture} 
\end{center}

\end{proposition}

\begin{proposition}{22}%Proposition I.22

\newparsen{
>Ek tri~wn e>ujei~wn,\\
a<'i e>isin >'isai\\
tris`i ta~ic doje'isaic%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} [e>uje'iaic],\\
tr'igwnon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal};\\
de~i%
\eix{\gr{de~i}} d`e\footnotemark\\
t`ac d'uo t~hc loip~hc
me'izonac e>~inai\\
p'anth| metalambanom'enac%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp}\\
{}[di`a\eix{\gr{di'a}} t`o \kai%
\eix{\gr{ka`i}} pant`oc trig'wnou\\
t`ac d'uo pleur`ac\\
t~hc loip~hc me'izonac e>~inai\\
p'anth| metalambanom'enac%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp}].
}
{
\"U\c c\ do\u grudan,\\
e\c sit olan\\
verilmi\c s\ \"u\c c\ do\u gruya,\\
bir \"u\c cgen in\c sa etmek;\\
ama \dei\\
ikisinin, \kalan dan
b\"uy\"uk olmas\i,\\
\nasilalinirsa,\\
\dia\ herhangi bir \"u\c cgenin,\\
iki kenar\i\\
\kalan dan b\"uy\"ukt\"ur,\\
\nasilalinirsa.
}
\footnotetext{Heiberg'e g\"ore \cite{Euclid-Heiberg}, Proklus'un \cite{MR1200456} ve Eutokios'un a\c c\i klamar\i n\i n metinlerinde \gr{d'e} yaz\i l\i r; ama \"Oklid'in metinlerinde \gr{d'h} yaz\i l\i r.}

\newparsen{
>'Estwsan\\
a<i doje~isai%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} tre~ic e>uje~iai
a<i A, B, G,\\
<~wn a<i d'uo t~hc loip~hc\\
me'izonec >'estwsan\\
p'anth| metalamban'omenai%
\eix{\gr{lamb'anw}!\gr{metalamb'anw}!\gr{metalamban'omenoc} part pres mp},\\
a<i m`en%
\eix{\gr{m'en}} A, B t~hc G,\\
a<i d`e A, G t~hc B,\\
\kai%
\eix{\gr{ka`i}} >'eti%
\eix{\gr{>'eti}} 
a<i B, G t~hc A;
}
{
Olsun\\
\"u\c c verilmi\c s do\u gru
\gk A, \gk B, ve \gk G,\\
ve ikisi, kalandan\\
b\"uy\"uk olsun,\\
\nasilalinirsa:\\
\gk A ile \gk B, \gk G'dan,\\
\gk A ile \gk G, \gk B'dan,\\
ve \gk B ile \gk G, \gk A'dan. 
}

\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
{}>ek t~wn >'iswn ta~ic A, B, G\\
tr'igwnon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}.
}
{
\Deidee\\
 \gk A, \gk B ve \gk G'ya e\c sit olanlardan\\
bir \"u\c cgen in\c sa etmek.
}

\newparsen{
>Ekke'isjw%
\eix{\gr{ke~imai}!\gr{>ekke~imai}!\gr{>ekke'isjw} 3rd sg pres imperat mp}\\
tic e>uje~ia <h DE\\
peperasm'enh%
\eix{\gr{pera'inw}!\gr{peperasm'enoc} part perf mp} m`en%
\eix{\gr{m'en}} kat`a t`o D\\
{}>'apeiroc d`e kat`a t`o E,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| m`en%
\eix{\gr{m'en}} A >'ish <h DZ,\\
t~h| d`e B >'ish <h ZH,\\
t~h| d`e G >'ish <h HJ;\\
\kai%
\eix{\gr{ka`i}} k'entrw| m`en%
\eix{\gr{m'en}} t~w| Z,\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| ZD\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o DKL;\\
p'alin\eix{\gr{p'alin}}
k'entrw| m`en%
\eix{\gr{m'en}} t~w| H,\\
diast'hmati%
\eix{\gr{<'isthmi}!\gr{di-'isthmi}!\gr{di'asthma}} d`e t~w| HJ\\
k'ukloc gegr'afjw%
\eix{\gr{gr'afw}!\gr{gegr'afjw} perf imperat mp} <o KLJ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp} a<i KZ, KH;
}
{
Oturtulsun\\
bir \gk{DE} do\u grusu,\\
\gk D'da s\i n\i rlanm\i\c s,\\
ama \gk E'da s\i n\i rlanmam\i\c s,\\
ve otursun\\
\gk A'ya e\c sit \gk{DZ},\\
\gk B'ya e\c sit \gk{ZH},\\
ve \gk G'ya e\c sit  \gk{HJ};\\
ve \gk Z merkezine\\
\gk{ZD} uzakl\i \u g\i nda \\
bir \gk{DKL} dairesi \c cizilmi\c s\ olsun;\\
\palin,
\gk H merkezine,\\
\gk{HJ} uzakl\i \u g\i nda,\\
\gk{KLJ} dairesi \c cizilmi\c s\ olsun,\\
ve \gk{KZ} ile \gk{KH} birle\c stirilmi\c s\ olsun.
}

\newparsen{
l'egw, <'oti\\
{}>ek tri~wn e>ujei~wn\\
t~wn >'iswn ta~ic A, B, G\\
tr'igwnon sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp} t`o KZH.
}
{
\Legohoti\\
\"u\c c\ do\u grudan\\
\gk A, \gk B ve \gk G'ya e\c sit olan\\
\gk{KZH} \"u\c cgeni in\c sa edilmi\c stir.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} g`ar t`o Z shme~ion
k'entron >est`i to~u DKL k'uklou,\\
{}>'ish >est`in <h ZD t~h| ZK;\\
{}>all`a%
\eix{\gr{>all'a}} <h ZD t~h| A >estin >'ish.\\
\kai%
\eix{\gr{ka`i}} <h KZ >'ara\eix{\gr{>'ara}} t~h| A >estin >'ish.\\
p'alin\eix{\gr{p'alin}},
{}>epe`i%
\eix{\gr{>epe'i}} t`o H shme~ion
k'entron >est`i to~u LKJ k'uklou,\\
{}>'ish >est`in <h HJ t~h| HK;\\
{}>all`a%
\eix{\gr{>all'a}} <h HJ t~h| G >estin >'ish;\\
\kai%
\eix{\gr{ka`i}} <h KH >'ara\eix{\gr{>'ara}} t~h| G >estin >'ish.\\
{}>est`i d`e \kai%
\eix{\gr{ka`i}} <h ZH t~h| B >'ish;\\
a<i tre~ic >'ara\eix{\gr{>'ara}} e>uje~iai\\
a<i KZ, ZH, HK\\
tris`i ta~ic A, B, G >'isai e>is'in.
}
{
\Gar, \gk Z noktas\i,
\gk{DKL} dairesinin merkezi oldu\u gundan,\\
\gk{ZD}, \gk{ZK}'ya e\c sittir;\\
ama \gk{ZD}, \gk A'ya e\c sittir.\\
Ve \gk{KZ} \ara\ \gk A'ya e\c sittir.\\
\Palin, \gk H noktas\i,
\gk{LKJ} dairesinin merkezi oldu\u gundan,\\
\gk{HJ}, \gk{HK} do\u grusuna e\c sittir;\\
ama \gk{HJ}, \gk G'ya e\c sittir;\\
ve \gk{KH} \ara\ \gk G'ya e\c sittir.\\
ve \gk{ZH}, \gk B do\u grusuna e\c sittir;\\
\ara\ \"u\c c\ do\u gru,\\
\gk{KZ}, \gk{ZH} ve \gk{HK},\\
\gk A, \gk B ve \gk G \"u\c cl\"us\"une e\c sittir.
}

\newparsen{
>Ek tri~wn >'ara\eix{\gr{>'ara}} e>ujei~wn\\
t~wn KZ, ZH, HK,\\
a<'i e>isin >'isai\\
tris`i ta~ic doje'isaic%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'iaic\\
ta~ic A, B, G,\\
tr'igwnon sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp} t`o KZH;\\
<'oper >'edei poi~hsai.
}
{
\Ara,  \"u\c c\ do\u grudan,\\
\gk{KZ}, \gk{ZH} ve \gk{HK}'dan,\\
e\c sit olan\\
verilmi\c s\ \"u\c c\ do\u gruya\\
\gk A, \gk B ve \gk G'ya,\\
bir \gk{KZH} \"u\c cgeni in\c sa edilmi\c stir;\\
\ozqed.
}

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\end{proposition}

\begin{proposition}{23}%Proposition I.23

\newparsen{
Pr`oc t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw|\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'ammw| >'ishn\\
gwn'ian e>uj'ugrammon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}.
}
{
Verilmi\c s\ bir do\u gruda,\\
ve \"uzerinde verilmi\c s\ noktada,\\
verilmi\c s\ d\"uzkenar a\c c\i ya e\c sit olan,\\
bir d\"uzkenar a\c c\i\ in\c sa etmek.
}

\newparsen{
>'Estw\\
<h m`en%
\eix{\gr{m'en}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h AB,\\
t`o d`e pr`oc a>ut~h| shme~ion t`o A,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc\\
<h <up`o DGE;
}
{
Olsun\\
verilmi\c s do\u gru \gk{AB},\\
ve \"uzerindeki nokta \gk A,\\
ve verilmi\c s d\"uzkenar a\c c\i\\
\gk{DGE}.
}

\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
pr`oc t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| AB\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| A\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'ammw|
t~h| <up`o DGE
{}>'ishn\\
gwn'ian e>uj'ugrammon\\
sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}.
}
{
\Deidee,\\
verilmi\c s\ \gk{AB} do\u grusunda,\\
ve \"uzerindeki \gk A noktas\i nda,\\
verilmi\c s\ d\"uzkenar
\gk{DGE} a\c c\i s\i na
e\c sit olan\\
bir d\"uzkenar a\c c\i\\
in\c sa etmek.
}

\newparsen{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp}\\
{}>ef> <ekat'erac t~wn GD, GE\\
tuq'onta shme~ia t`a D, E,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h DE;\\
\kai%
\eix{\gr{ka`i}} >ek tri~wn e>ujei~wn,\\
a<'i e>isin >'isai tris`i
ta~ic GD, DE, GE,\\
tr'igwnon sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp} t`o AZH,\\
<'wste%
\eix{\gr{<'wste}} >'ishn e>~inai\\
t`hn m`en%
\eix{\gr{m'en}} GD t~h| AZ,\\
t`hn d`e GE t~h| AH,\\
\kai%
\eix{\gr{ka`i}} >'eti%
\eix{\gr{>'eti}} 
t`hn DE t~h| ZH.
}
{
al\i nm\i\c s olsun\\
\gk{GD} ve \gk{GE}'un her birinden\\
rastgele \gk D ve \gk E noktalar\i,\\
ve \gk{DE} birle\c stirilmi\c s\ olsun,\\
ve \"u\c c\ do\u grudan\\
\"u\c c \gk{GD}, \gk{DE} ve \gk{GE}'a
e\c sit olan,\\
\gk{AZH} \"u\c cgeni in\c sa edilmi\c s\ olsun\\
\hosteB\ e\c sit olsun\\
\gk{GD}, \gk{AZ}'ya,\\
\gk{GE}, \gk{AH}'ya,\\
ve ayr\i ca \gk{DE}, \gk{ZH}'ya.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} d'uo a<i DG, GE\\
d'uo ta~ic ZA, AH >'isai e>is`in\\
<ekat'era <ekat'era|,\\
\kai%
\eix{\gr{ka`i}} b'asic <h DE\\
b'asei t~h| ZH >'ish,\\
gwn'ia >'ara\eix{\gr{>'ara}} <h <up`o DGE gwn'ia|\\
t~h| <up`o ZAH >estin >'ish.
}
{
\Oun\ \gk{DG} ve \gk{GE} ikilisi,\\
\gk{ZA} ve \gk{AH} ikilisine e\c sit oldu\u gundan,\\
her biri birine,\\
ve \gk{DE} taban\i,\\
\gk{ZH} taban\i na
e\c sit oldu\u gundan,\\
\ara\ \gk{DGE} a\c c\i s\i\\
\gk{ZAH}'ya e\c sittir. 
}

\newparsen{
Pr`oc >'ara\eix{\gr{>'ara}} t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| AB\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| A\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'ammw| t~h| <up`o DGE >'ish\\
gwn'ia e>uj'ugrammoc sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp} <h <up`o ZAH;\\
<'oper >'edei poi~hsai.\\
}
{
\Ara, verilmi\c s \gk{AB} do\u grusunda,\\
ve  \"uzerindeki \gk A noktas\i nda,\\
verilmi\c s d\"uzkenar \gk{DGE} a\c c\i s\i na e\c sit olan\\
\gk{ZAH} d\"uzkenar a\c c\i s\i\ in\c sa edilmi\c stir;\lli\\
\ozqef.
}
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}
\end{center}
\end{proposition}

\begin{proposition}{24}%Proposition I.24

\newparsen{
>E`an d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
{}[ta~ic] d'uo pleura~ic >'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|,\\
t`hn d`e gwn'ian\\
t~hc gwn'iac me'izona >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp},\\
\kai%
\eix{\gr{ka`i}} t`hn b'asin\\
t~hc b'asewc me'izona <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}.
}
{
E\u ger iki \"u\c cgende\\
iki kenar\\
iki kenara e\c sitse,\\
her biri birine,\\
ama a\c c\i\\
a\c c\i dan b\"uy\"ukse,\\
{}[yani] e\c sit kenarlarca\\
rapteden,\\
taban da\\
tabandan b\"uy\"uk olacak.
}

\begin{center}
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\usefont{T1}{ptm}{m}{n}
\rput(0.10828125,-1.336875){\gk G}
\usefont{T1}{ptm}{m}{n}
\rput(4.7471876,1.383125){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(5.709219,-0.756875){\gk E}
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\usefont{T1}{ptm}{m}{n}
\rput(4.1303124,-1.416875){\gk Z}
\end{pspicture} 
\end{center}


\newparsen{
>'Estw\\
d'uo tr'igwna t`a ABG, DEZ\\
t`ac d'uo pleur`ac t`ac AB, AG\\
ta~ic d'uo pleura~ic ta~ic DE, DZ\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
<ekat'eran <ekat'era|,\\
t`hn m`en%
\eix{\gr{m'en}} AB t~h| DE\\
t`hn d`e AG t~h| DZ,\\
<h d`e pr`oc t~w| A gwn'ia\\
t~hc pr`oc t~w| D gwn'iac
me'izwn >'estw;
}
{
Olsun\\
iki \"u\c cgen \gk{ABG} ve \gk{DEZ},\\
iki \gk{AB} ve \gk{AG} kenar\i,\\
iki \gk{DE} ve \gk{DZ} kenar\i na\\
e\c sit olan,\\
her biri birine,\\
\gk{AB}, \gk{DE}'a,\\
ve \gk{AG}, \gk{DZ}'ya,\\
ve \gk A'daki a\c c\i,\\
\gk D'daki a\c c\i s\i ndan b\"uy\"uk olsun.
}

\newparsen{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} b'asic <h BG\\
b'asewc t~hc EZ
me'izwn >est'in.
}
{
\Legohoti\\
 \gk{BG} taban\i\ da\\
 \gk{EZ} taban\i ndan b\"uy\"ukt\"ur.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} g`ar me'izwn
<h <up`o BAG gwn'ia\\
t~hc <up`o EDZ gwn'iac,\\
sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
pr`oc t~h| DE e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| D\\
t~h| <up`o BAG gwn'ia| >'ish <h <up`o EDH,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
<opot'era| t~wn AG, DZ >'ish
<h DH,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp}
a<i EH, ZH.
}
{
\Gar\ \gk{BAG} a\c c\i s\i, b\"uy\"uk oldu\u gundan\\
\gk{EDZ} a\c c\i s\i ndan,\\
in\c sa edilmi\c s\ olsun\\
\gk{DE} do\u grusunda,\\
ve \"uzerindeki \gk D noktas\i nda,\\
\gk{BAG} a\c c\i s\i na e\c sit olan
\gk{EDH},\\
ve oturmu\c s olsun\\
\gk{AG} veya \gk{DZ}'ya e\c sit olan
\gk{DH},\\
ve \gk{EH} ve \gk{ZH} birle\c stirilmi\c s olsun.
}

\newparsen{
{}>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} AB t~h| DE,\\
<h d`e AG t~h| DH,\\
d'uo d`h%
\eix{\gr{d'h}} a<i BA, AG\\
dus`i ta~ic ED, DH >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o BAG\\
gwn'ia| t~h| <up`o EDH >'ish;\\
b'asic >'ara\eix{\gr{>'ara}} <h BG\\
b'asei t~h| EH >estin >'ish.\\
p'alin\eix{\gr{p'alin}},
{}>epe`i%
\eix{\gr{>epe'i}} >'ish >est`in
<h DZ t~h| DH,\\
{}>'ish >est`i \kai%
\eix{\gr{ka`i}}\\
<h <up`o DHZ gwn'ia t~h| <up`o DZH;\\
me'izwn >'ara\eix{\gr{>'ara}}\\
<h <up`o DZH t~hc <up`o EHZ;\\
poll~w| >'ara\eix{\gr{>'ara}} me'izwn >est`in\\
<h <up`o EZH t~hc <up`o EHZ.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} tr'igwn'on >esti t`o EZH\\
me'izona >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
t`hn <up`o EZH gwn'ian t~hc <up`o EHZ,\\
<up`o d`e t`hn me'izona gwn'ian\\
<h me'izwn pleur`a\\
<upote'inei%
\eix{\gr{te'inw}!\gr{<upote'inw}},\\
me'izwn >'ara\eix{\gr{>'ara}} \kai%
\eix{\gr{ka`i}}\\
pleur`a <h EH t~hc EZ.\\
{}>'ish d`e <h EH t~h| BG;\\
me'izwn >'ara\eix{\gr{>'ara}} \kai%
\eix{\gr{ka`i}} <h BG t~hc EZ.
}
{
\Oun\ e\c sit oldu\u gundan\\
\gk{AB}, \gk{DE}'a,\\
ve \gk{AG}, \gk{DH}'ya,\\
\dee\ \gk{BA} ve \gk{AG} ikilisi,\\
\gk{ED} ve \gk{DH} iklisine
e\c sittir,\\
her biri birine;\\
ve \gk{BAG} a\c c\i s\i\\
\gk{EDH}  a\c c\i s\i na e\c sittir;\\
\ara\ \gk{BG}   taban\i\\
\gk{EH} taban\i na e\c sittir.\\
\Palin,
\gk{DZ}, \gk{DH}'ya e\c sit oldu\u gundan,\\
bir de e\c sittir\\
\gk{DHZ} a\c c\i s\i, \gk{DZH}'ya;\\
\ara\ b\"uy\"ukt\"ur\\
\gk{DZH}, \gk{EHZ}'dan;\\
\ara\ \c cok daha b\"uy\"ukt\"ur\\
\gk{EZH}, \gk{EHZ} a\c c\i s\i ndan.\\
Ve \gk{EZH} \"u\c cgende,\\
b\"uy\"uk oldu\u gundan\\
\gk{EZH} a\c c\i s\i\ \gk{EHZ}'dan,\\
ve daha b\"uy\"uk a\c c\i,\\
daha b\"uy\"uk a\c c\i\ taraf\i ndan\\
raptedildi\u ginden,\\
\ara\ b\"uy\"ukt\"ur\\
\gk{EH} kenar\i\ da \gk{EZ}'dan.\\
Ve \gk{EH}, \gk{BG}'ya e\c sittir;\\
\ara\ \gk{BG} da, \gk{EZ}'dan b\"uy\"ukt\"ur.
}

\newparsen{
>E`an >'ara\eix{\gr{>'ara}} d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
dus`i pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|,\\
t`hn d`e gwn'ian\\
t~hc gwn'iac
me'izona >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp},\\
\kai%
\eix{\gr{ka`i}} t`hn b'asin\\
t~hc b'asewc
me'izona <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  iki \"u\c cgende\\
iki kenar\\
iki kenara e\c sitse \\
her biri birine,\\
ama a\c c\i\\
a\c c\i dan b\"uy\"ukse,\\
{}[yani] e\c sit kenarlarca\\
rapteden,\\
taban da\\
tabandan b\"uy\"uk olacak;\\
\ozqed.
}

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\rput(4.1303124,-1.416875){\gk Z}
\end{pspicture} 
\end{center}

\end{proposition}

\begin{proposition}{25}%Proposition I.25

\newparsen{
>E`an d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
dus`i pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|,\\
t`hn d`e b'asin\\
t~hc b'asewc
me'izona >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act},\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian\\
t~hc gwn'iac
me'izona <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp}.
}
{
E\u ger iki \"u\c cgende\\
iki kenar\\
iki kenara
e\c sitse\\
her biri birine,\\
ama taban\\
tabandan b\"uy\"ukse,\\
a\c c\i\ da\\
a\c c\i dan
b\"uy\"uk olacak\\
{}[yani] e\c sit do\u grularca\\
rapteden.
}

\newparsen{
>'Estw\\
d'uo tr'igwna t`a ABG, DEZ\\
t`ac d'uo pleur`ac t`ac AB, AG\\
ta~ic d'uo pleura~ic ta~ic DE, DZ\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
<ekat'eran <ekat'era|,\\
t`hn m`en%
\eix{\gr{m'en}} AB t~h| DE,\\
t`hn d`e AG t~h| DZ;\\
b'asic d`e <h BG\\
b'asewc t~hc EZ
me'izwn >'estw;
}
{
Olsun\\
iki \"u\c cgen \gk{ABG} ve \gk{DEZ},\\
iki \gk{AB} ve \gk{AG} kenar\i,\\
iki \gk{DE} ve \gk{DZ} kenar\i na\\
e\c sit olan,\\
her biri birine,\\
\gk{AB}, \gk{DE}'a\\
ve \gk{AG}, \gk{DZ}'ya;\\
ve \gk{BG} taban\i\\
 \gk{EZ} taban\i ndan b\"uy\"uk olsun.
}

\newparsen{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o BAG\\
gwn'iac t~hc <up`o EDZ
me'izwn >est'in.
}
{
\Legohoti\\
\gk{BAG} a\c c\i s\i\ da\\
\gk{EDZ} a\c c\i s\i ndan
b\"uy\"ukt\"ur.
}

\newparsen{
E>i g`ar m'h,\\
{}>'htoi >'ish >est`in a>ut~h| >`h >el'asswn;\\
{}>'ish m`en%
\eix{\gr{m'en}} o>~un\eix{\gr{o>~un}} o>uk >'estin\\
<h <up`o BAG t~h| <up`o EDZ;\\
{}>'ish g`ar >`an >~hn\\
\kai%
\eix{\gr{ka`i}} b'asic <h BG b'asei t~h| EZ;\\
o>uk >'esti d'e.\\
o>uk >'ara\eix{\gr{>'ara}} >'ish >est`i\\
gwn'ia <h <up`o BAG t~h| <up`o EDZ;\\
o>ud`e m`hn%
\eix{\gr{m'hn}} >el'asswn >est`in\\
<h <up`o BAG t~hc <up`o EDZ;\\
{}>el'asswn g`ar >`an >~hn\\
\kai%
\eix{\gr{ka`i}} b'asic <h BG b'asewc t~hc EZ;\lli\\
o>uk >'esti d'e;\\
o>uk >'ara\eix{\gr{>'ara}} >el'asswn >est`in\\
<h <up`o BAG gwn'ia t~hc <up`o EDZ.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d'e, <'oti\\
o>ud`e >'ish;\\
me'izwn >'ara\eix{\gr{>'ara}} >est`in\\
<h <up`o BAG t~hc <up`o EDZ.
}
{
\Gar\ e\u ger de\u gilse,\\
ya ona e\c sittir, ya da ondan k\"u\c c\"uk;\\
ama \oun\ e\c sit de\u gildir\\
\gk{BAG}, \gk{EDZ}'ya;\\
\gar \ e\u ger e\c sit ise\\
\gk{BG} taban\i\ da, \gk{EZ} taban\i na [e\c sittir];\\
ama de\u gil.\\
\Ara\ e\c sit de\u gildir\\
\gk{BAG} a\c c\i s\i,  \gk{EDZ}'ya;\\
\meen\ k\"u\c c\"uk de\u gildir\\
\gk{BAG}, \gk{EDZ}'dan;\\
\gar\ e\u ger k\"u\c c\"uk ise,\\
\gk{BG} taban\i\ da, \gk{EZ} taban\i ndan [k\"u\c c\"ukt\"ur];\\
ama de\u gil;\\
\ara\ k\"u\c c\"uk de\u gildir\\
\gk{BAG}, \gk{EDZ}'dan.\\
Ama g\"osterilmi\c sti ki\\
e\c sit de\u gildir;\\
\ara\ b\"uy\"ukt\"ur\\
\gk{BAG}, \gk{EDZ}'dan.
}

\newparsen{
{}>E`an >'ara\eix{\gr{>'ara}} d'uo tr'igwna\\
t`ac d'uo pleur`ac\\
dus`i pleura~ic
{}>'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ek'atera|,\\
t`hn d`e bas'in\\
t~hc b'asewc
me'izona >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act},\\
\kai%
\eix{\gr{ka`i}} t`hn gwn'ian\\
t~hc gwn'iac
me'izona <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
t`hn <up`o t~wn >'iswn e>ujei~wn\\
perieqom'enhn%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp};\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  iki \"u\c cgende\\
iki kenar\\
iki kenara
e\c sitse\\
her biri birine,\\
ama taban\\
tabandan
b\"uy\"ukse,\\
a\c c\i\ da\\
a\c c\i dan
b\"uy\"uk olacak\\
{}[yani] e\c sit do\u grularca\\
rapteden;\\
\ozqed.
}
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\rput(3.2151563,-1.216875){\gk G}
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(7.596719,-1.236875){\gk Z}
\end{pspicture} 
\end{center}
\end{proposition}

\begin{proposition}{26}%Proposition I.26
\parsen{
If two triangles\\
two angles\\
to two angles\\
have equal,\\
either to either,\\
and one side\\
to one side\\
equal,\\
either that near the equal sides\\
or that subtending\\
one of the equal sides,\\
also the remaining sides\\
to the remaining sides\\
they will have equal,\\
%either to either,\\
also the remaining angle\\
to the remaining angle.
}
{
{}>E`an d'uo tr'igwna\\
t`ac d'uo gwn'iac\\
dus`i gwn'iaic >'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} m'ian pleur`an\\
mi~a| pleur~a| >'ishn\\
{}>'htoi t`hn pr`oc ta~ic >'isaic gwn'iaic\\
{}>`h t`hn <upote'inousan%
\eix{\gr{te'inw}!\gr{<upote'inw}!\gr{<upote'inwn} part pres act}\\
<up`o m'ian t~wn >'iswn gwni~wn,\\
\kai%
\eix{\gr{ka`i}} t`ac loip`ac pleur`ac\\
ta~ic loipa~ic pleura~ic >'isac <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
%{}[<ekat'eran <ekat'era|]\\
\kai%
\eix{\gr{ka`i}} t`hn loip`hn gwn'ian\\
t~h| loip~h| gwn'ia|.
}
{
E\u ger iki \"u\c cgenin\\
iki a\c c\i s\i,\\
iki a\c c\i s\i na e\c sitse,\\
her biri birine,\\
ve bir kenar,\\
bir kenara e\c sitse,\\
ya e\c sit a\c c\i lar\i n aras\i nda olan \\
ya da kar\c s\i layan\\
e\c sit a\c c\i lardan birini,\\
kalan kenarlar\i\ da\\
kalan kenarlar\i na e\c sit olacak,\\
kalan a\c c\i lar\i\ da\\
kalan a\c c\i lar\i na.
}

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\parsen{
Let there be\\
two triangles, \gk{ABG} and \gk{DEZ}\\
the two angles \gk{ABG} and \gk{BGA}\\
to the two angles \gk{DEZ} and \gk{EZD}\\
having equal,\\
either to either,\\
\gk{ABG} to \gk{DEZ},\\
and \gk{BGA} to \gk{EZD};\\
and let them also have\\
one side\\
to one side\\
equal,\\
first that near the equal angles,\\
\gk{BG} to \gk{EZ};
}
{
{}>'Estw\\
d'uo tr'igwna t`a ABG, DEZ\\
t`ac d'uo gwn'iac t`ac <up`o ABG, BGA\\
dus`i ta~ic <up`o DEZ, EZD\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
<ekat'eran <ekat'era|,\\
t`hn m`en%
\eix{\gr{m'en}} <up`o ABG t~h| <up`o DEZ,\\
t`hn d`e <up`o BGA t~h| <up`o EZD;\\
{}>eq'etw d`e\\
\kai%
\eix{\gr{ka`i}} m'ian pleur`an\\
mi~a| pleur~a| >'ishn,\\
pr'oteron t`hn pr`oc ta~ic >'isaic gwn'iaic\\
t`hn BG t~h| EZ;}
{
Olsun\\
iki \"u\c cgen \gk{ABG} ve \gk{DEZ},\\
iki \gk{ABG} ve \gk{BGA} a\c c\i lar\i\\
iki \gk{DEZ} ve \gk{EZD}'ya\\
e\c sit olan,\\
her biri birine,\\
\gk{ABG}, \gk{DEZ}'ya\\
ve \gk{BGA}, \gk{EZD}'ya;\\
ayr\i ca olsun\\
bir kenar\i\ da\\
bir kenar\i na e\c sit,\\
\"once, esit a\c c\i lar\i n aras\i nda olan,\\
\gk{BG}, \gk{EZ}'ya.
}

\parsen{
I say that\\
the remaining sides\\
to the remaining sides\\
they will have equal,\\
either to either,\\
\gk{AB} to \gk{DE}\\
and \gk{AG} to \gk{DZ},\\
also the remaining angle\\
to the remaining angle,\\
\gk{BAG} to \gk{EDZ}.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} t`ac loip`ac pleur`ac\\
ta~ic loipa~ic pleura~ic >'isac <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
<ekat'eran <ekat'era|,\\
t`hn m`en%
\eix{\gr{m'en}} AB t~h| DE\\
t`hn d`e AG t~h| DZ,\\
\kai%
\eix{\gr{ka`i}} t`hn loip`hn gwn'ian\\
t~h| loip~h| gwn'ia|,\\
t`hn <up`o BAG t~h| <up`o EDZ.
}
{
\Legohoti\\
kalan kenarlar da\\
kalan kenarlara e\c sit olacaklar,\\
her biri birine,\\
\gk{AB}, \gk{DE}'a\\
ve \gk{AG}, \gk{DZ}'ya,\\
ve kalan a\c c\i\\
kalan a\c c\i ya,\\
\gk{BAG}, \gk{EDZ}'ya.
}

\parsen{
For, if it is unequal,\\
\gk{AB} to \gk{DE},\\
one of them is greater.\\
Let be greater\\
\gk{AB},\\
and let there be cut\\
to \gk{DE} equal\\
\gk{BH},\\
and suppose there has been joined\\
\gk{HG}.
}
{
E>i g`ar >'anis'oc >estin\\
<h AB t~h| DE,\\
m'ia a>ut~wn me'izwn >est'in.\\
{}>'estw me'izwn <h AB,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| DE >'ish <h BH,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw
<h HG.
}
{
\Gar\ e\u ger e\c sit de\u gilse,\\
\gk{AB}, \gk{DE} kenar\i na,\\
biri daha b\"uy\"ukt\"ur.\\
\gk{AB} daha b\"uy\"uk olsun,\\
ve oturmu\c s\ olsun\\
\gk{DE}'a e\c sit olan \gk{BH},\\
ve \gk{HG} birle\c stirilmi\c s\ olsun.
}


\parsen{
Because then it is equal,\\
\gk{BH} to \gk{DE},\\
and \gk{BG} to \gk{EZ},\\
the two, \gk{BH}\footnotemark\ and \gk{BG}\\
to the two \gk{DE} and \gk{EZ}\\
are equal,\\
either to either,\\
and the angle \gk{HBG}\\
to the angle \gk{DEZ}\\
is equal;\\
therefore the base \gk{HG}\\
to the base \gk{DZ}\\
is equal,\\
and the triangle \gk{HBG}\\
to the triangle \gk{DEZ}\\
is equal,\\
and the remaining angles\\
to the remaining angles\\
will be equal,\\
those that the equal sides subtend.\\
Equal therefore is angle \gk{BGH}\\
to \gk{DZE}.\\
But \gk{DZE}\\
to \gk{BGA}\\
is supposed equal;\\
therefore also \gk{BGH}\\
to \gk{BGA}\\
is equal,\\
the lesser to the greater,\\
which is impossible.\\
Therefore it is not unequal,\\
\gk{AB} to \gk{DE}.\\
Therefore it is equal.\\
It is also the case that\\
\gk{BG} to \gk{EZ} is equal;\\
then the two \gk{AB} and \gk{BG}\\
to the two \gk{DE} and \gk{EZ}\\
are equal,\\
either to either;\\
also the angle \gk{ABG}\\
to the angle \gk{DEZ}\\
is equal;\\
therefore the base \gk{AG}\\
to the base \gk{DZ}\\
is equal,\\
and the remaining angle \gk{BAG}\\
to the remaining angle \gk{EDZ}\\
is equal.
}
{
{}>Epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} BH t~h| DE,\\
<h d`e BG t~h| EZ,\\
d'uo d`h%
\eix{\gr{d'h}} a<i BH, BG\\
dus`i ta~ic DE, EZ >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o HBG\\
gwn'ia| t~h| <up`o DEZ >'ish >est'in;\\
b'asic >'ara\eix{\gr{>'ara}} <h HG\\
b'asei t~h| DZ >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o HBG tr'igwnon\\
t~w| DEZ trig'wnw| >'ison >est'in,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai,\\
<uf> <`ac a<i >'isai pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
{}>'ish >'ara\eix{\gr{>'ara}} <h <up`o HGB gwn'ia\\
t~h| <up`o DZE.\\
{}>all`a%
\eix{\gr{>all'a}} <h <up`o DZE\\
t~h| <up`o BGA <up'okeitai%
\eix{\gr{ke~imai}!\gr{<up'okeimai}} >'ish;\\
\kai%
\eix{\gr{ka`i}} <h <up`o BGH >'ara\eix{\gr{>'ara}}\\
t~h| <up`o BGA >'ish >est'in,\\
<h >el'asswn t~h| me'izoni;\\
<'oper >ad'unaton\eix{\gr{<'oper >ad'unaton}}.\\
o>uk >'ara\eix{\gr{>'ara}} >'anis'oc >estin\\
<h AB t~h| DE.\\
{}>'ish >'ara\eix{\gr{>'ara}}.\\
{}>'esti d`e \kai%
\eix{\gr{ka`i}}\\
<h BG t~h| EZ >'ish;\\
d'uo d`h%
\eix{\gr{d'h}} a<i AB, BG\\
dus`i ta~ic DE, EZ >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ABG\\
gwn'ia| t~h| <up`o DEZ >estin >'ish;\\
b'asic >'ara\eix{\gr{>'ara}} <h AG\\
b'asei t~h| DZ >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} loip`h gwn'ia <h <up`o BAG\\
t~h| loip~h| gwn'ia| t~h| <up`o EDZ\\
{}>'ish >est'in.
}
{
\Oun\ e\c sit oldu\u gundan\\
\gk{BH}, \gk{DE}'a\\
ve \gk{BG}, \gk{EZ}'ya,\\
\dee\ \gk{BH} ve \gk{BG} ikilisi\\
\gk{DE} ve \gk{EZ} ikilisine e\c sittir,\\
her biri birine,\\
ve \gk{HBG} a\c c\i s\i\\
\gk{DEZ} a\c c\i s\i na e\c sittir;\\
\ara\ \gk{HG} taban\i\\
\gk{DZ} taban\i na e\c sittir,\\
ve \gk{HBG} \"u\c cgeni\\
\gk{DEZ} \"u\c cgenine e\c sittir,\\
ve kalan a\c c\i lar\\
kalan a\c c\i lara e\c sit olacaklar\\
e\c sit kenarlar raptetti\u gi.\\
\Ara\ \gk{BGH} a\c c\i s\i\ e\c sittir\\
\gk{DZE}'a.\\
Ama \gk{DZE},\\
\gk{BGA}'ya e\c sit kabul edilir,\\
\ara\  \gk{BGH} de\\
\gk{BGA} a\c c\i s\i na e\c sittir,\\
daha k\"u\c c\"uk olan daha b\"uy\"uk olana,\\
\imkansiz.\\
\Ara\ e\c sit de\u gil de\u gildir,\\
\gk{AB}, \gk{DE} kenar\i na.\\
\Ara\ e\c sittir.\\
Ve durum \c s\"oyledir;\\
\gk{BG}, \gk{EZ} kenar\i na e\c sittir;\\
\dee\ \gk{AB} ve \gk{BG} ikilisi\\
 \gk{DE} ve \gk{EZ} ikilisine e\c sittir,\\
her biri birine;\\
\gk{ABG} a\c c\i s\i\ da\\
\gk{DEZ} a\c c\i s\i na e\c sittir;\\
\ara\ \gk{AG} taban\i\\
\gk{DZ} taban\i na e\c sittir,\\
ve kalan \gk{BAG} a\c c\i s\i\\
kalan \gk{EDZ} a\c c\i s\i na\\
e\c sittir.
}

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\parsen{
But then again let them be\\
---[those angles] equal sides\\
subtending---\\
equal,\\
as \gk{AB} to \gk{DE};\\
I say again that\\
also the remaining sides\\
to the remaining sides\\
will be equal,\\
\gk{AG} to \gk{DZ},\\
and \gk{BG} to \gk{EZ},\\
and also the remaining angle \gk{BAG}\\
to the remaining angle \gk{EDZ}\\
is equal. 
}
{
>all`a%
\eix{\gr{>all'a}} d`h%
\eix{\gr{d'h}} p'alin\eix{\gr{p'alin}} >'estwsan\\
a<i <up`o t`ac >'isac gwn'iac pleura`i <upote'inousai%
\eix{\gr{te'inw}!\gr{<upote'inw}!\gr{<upote'inwn} part pres act}
{}>'isai,\\
<wc <h AB t~h| DE;\\
l'egw p'alin\eix{\gr{p'alin}}, <'oti\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i pleura`i\\
ta~ic loipa~ic pleura~ic >'isai >'esontai,\\
<h m`en%
\eix{\gr{m'en}} AG t~h| DZ,\\
<h d`e BG t~h| EZ\\
\kai%
\eix{\gr{ka`i}} >'eti%
\eix{\gr{>'eti}} 
<h loip`h gwn'ia <h <up`o BAG\\
t~h| loip~h| gwn'ia| t~h| <up`o EDZ\\
{}>'ish >est'in.
}
{
Ama \dee\ \palin\ olsun\\
e\c sit a\c c\i lar\i\ rapteden kenarlar e\c sit,\lli\\
\gk{AB}, \gk{DE} kenar\i na gibi;\\
\Palin\ \legohoti\\
kalan kenarlar da\\
kalan kenarlara e\c sit olacaklar,\\
\gk{AG}, \gk{DZ} kenar\i na\\
ve \gk{BG}, \gk{EZ} kenar\i na\\
ve kalan \gk{BAG} a\c c\i s\i\ da\\
kalan \gk{EDZ} a\c c\i s\i na\\
e\c sittir. 
}

\parsen{
For, if it is unequal,\\
\gk{BG} to \gk{EZ},\\
one of them is greater.\\
Let be greater,\\
if possible,\\
\gk{BG},\\
and let there be cut\\
to \gk{EZ} equal\\
\gk{BJ},\\
and suppose there has been joined\\
\gk{AJ}.\\
Because also it is equal\\
---\gk{BJ} to \gk{EZ}\\
and \gk{AB} to \gk{DE},\\
then the two \gk{AB} and \gk{BJ}\\
to the two \gk{DE} and \gk{EZ}\\
are equal,\\
either to either;\\
and they contain equal angles;\\
therefore the base \gk{AJ}\\
to the base \gk{DZ}\\
is equal,\\
and the triangle \gk{ABJ}\\
to the triangle \gk{DEZ}\\
is equal,\\
and the remaining angles\\
to the remaining angles\\
are equal,\\
which the equal sides\\
subtend.\\
Therefore equal is\\
angle \gk{BJA}\\
to \gk{EZD}.\\
But \gk{EZD}\\
to \gk{BGA}\\
is equal;\\
then of triangle \gk{AJG}\\
the exterior angle \gk{BJA}\\
is equal\\
to the interior and opposite\\
\gk{BGA};\\
which is impossible.\\
Therefore it is not unequal,\\
\gk{BG} to \gk{EZ};\\
therefore it is equal.\\
And it is also,\\
\gk{AB},\\
to \gk{DE},\\
equal.\\
Then the two \gk{AB} and \gk{BG}\\
to the two \gk{DE} and \gk{EZ}\\
are equal,\\
either to either;\\
and equal angles\\
they contain;\\
therefore the base \gk{AG}\\
to the base \gk{DZ}\\
is equal,\\
and triangle \gk{ABG}\\
to triangle \gk{DEZ}\\
is equal,\\
and the remaining angle \gk{BAG}\\
to the remaining angle \gk{EDZ}\\
is equal.
}
{
E>i g`ar >'anis'oc >estin\\
<h BG t~h| EZ,\\
m'ia a>ut~wn me'izwn >est'in.\\
{}>'estw me'izwn, e>i dunat'on, <h BG,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| EZ >'ish <h BJ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw <h AJ.\\
\kai%
\eix{\gr{ka`i}} >ep`ei >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} BJ t~h| EZ\\
<h d`e AB t~h| DE,\\
d'uo d`h%
\eix{\gr{d'h}} a<i AB, BJ\\
dus`i ta~ic DE, EZ >'isai e>is`in\\
<ekat'era <ekar'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'iac >'isac peri'eqousin%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act};\\
b'asic >'ara\eix{\gr{>'ara}} <h AJ\\
b'asei t~h| DZ >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o ABJ tr'igwnon\\
t~w| DEZ trig'wnw| >'ison >est'in,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai,\\
<uf> <`ac a<i >'isac pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
{}>'ish >'ara\eix{\gr{>'ara}} >est`in\\
<h <up`o BJA gwn'ia t~h| <up`o EZD.\\
{}>all`a%
\eix{\gr{>all'a}} <h <up`o EZD\\
t~h| <up`o BGA >estin >'ish;\\
trig'wnou d`h%
\eix{\gr{d'h}} to~u AJG\\
<h >ekt`oc gwn'ia <h <up`o BJA >'ish >est`i\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion t~h| <up`o BGA;\\
<'oper >ad'unaton%
\eix{\gr{<'oper >ad'unaton}}.\\
o>uk >'ara\eix{\gr{>'ara}} >'anis'oc >estin\\
<h BG t~h| EZ;\\
{}>'ish >'ara\eix{\gr{>'ara}}.\\
{}>est`i d`e \kai%
\eix{\gr{ka`i}} <h AB t~h| DE >'ish.\\
d'uo d`h%
\eix{\gr{d'h}} a<i AB, BG\\
d'uo ta~ic DE, EZ >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'iac >'isac peri'eqousi%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{peri'eqwn} part pres act};\\
b'asic >'ara\eix{\gr{>'ara}} <h AG\\
b'asei t~h| DZ >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o ABG tr'igwnon\\
t~w| DEZ trig'wnw| >'ison\\
\kai%
\eix{\gr{ka`i}} loip`h gwn'ia <h <up`o BAG\\
t~h| loip`h| gwn'ia| t~h| <up`o EDZ >'ish.
}
{
\Gar\ e\u ger e\c sit de\u gil ise,\\
\gk{BG}, \gk{EZ} kenar\i na,\\
biri daha b\"uy\"ukt\"ur.\\
M\"umk\"unse, \gk{BG} daha b\"uy\"uk olsun,\\
ve oturmu\c s\ olsun\\
\gk{EZ}'ya e\c sit olan \gk{BJ},\\
ve \gk{AJ} birle\c stirilmi\c s\ olsun.\\
Ayr\i ca e\c sit oldu\u gundan\\
\gk{BJ},  \gk{EZ} kenar\i na,\\
ve \gk{AB}, \gk{DE} kenar\i na,\\
\dee\ \gk{AB} ve \gk{BJ} ikilisi,\\
\gk{DE} ve \gk{EZ} ikilisine e\c sittir,\\
her biri birine;\\
ve e\c sit a\c c\i lar\i\ i\c cerirler,\\
\ara\  \gk{AJ} taban\i\\
\gk{DZ} taban\i na e\c sittir,\\
ve \gk{ABJ} \"u\c cgeni\\
 \gk{DEZ} \"u\c cgenine e\c sittir,\\
ve kalan a\c c\i lar\\
kalan a\c c\i lara e\c sit olacak,\\
e\c sit kenarlar\i n raptetti\u gi.\\
\Ara\ e\c sittir\\
\gk{BJA} a\c c\i s\i, \gk{EZD} a\c c\i s\i na.\\
Ama \gk{EZD},\\
 \gk{BGA} a\c c\i s\i na e\c sittir;\\
o halde \gk{AJG} \"u\c cgeninin\\
\gk{BJA} d\i\c s\ a\c c\i s\i\ e\c sittir\\
i\c c\ ve kar\c s\i t \gk{BGA} a\c c\i s\i na;\\
\imkansiz.\\
\Ara\ e\c sit de\u gil de\u gildir\\
\gk{BG}, \gk{EZ}'ya;\\
\ara\ e\c sittir.\\
Ve tekrar \gk{AB}, \gk{DE} kenar\i na e\c sittir.\\
O halde \gk{AB} ve \gk{BG} ikilisi\\
 \gk{DE} ve \gk{EZ} ikilisine e\c sittir,\\
her biri birine;\\
ve e\c sit a\c c\i lar i\c cerirler;\\
\ara\ \gk{AG} taban\i\\
\gk{DZ} taban\i na e\c sittir,\\
ve  \gk{ABG} \"u\c cgeni\\
 \gk{DEZ} \"u\c cgenine e\c sittir,\\
ve kalan \gk{BAG} a\c c\i s\i\\
kalan \gk{EDZ} a\c c\i s\i na e\c sittir.
}

\parsen{
If therefore two triangles\\
two angles\\
to two angles\\
have equal,\\
either to either,\\
and one side\\
to one side\\
equal,\\
either that near the equal sides\\
or that subtending\\
one of the equal sides,\\
also the remaining sides\\
to the remaining sides\\
they will have equal,\\
%either to either,\\
also the remaining angle\\
to the remaining angle;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} d'uo tr'igwna\\
t`ac d'uo gwn'iac\\
dus`i gwn'iaic >'isac >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act}\\
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} m'ian pleur`an\\
mi~a| pleur~a| >'ishn\\
{}>'htoi t`hn pr`oc ta~ic >'isaic gwn'iaic,\\
{}>`h t`hn <upote'inousan%
\eix{\gr{te'inw}!\gr{<upote'inw}!\gr{<upote'inwn} part pres act}\\
<up`o m'ian t~wn >'iswn gwni~wn,\\
\kai%
\eix{\gr{ka`i}} t`ac loip`ac pleur`ac\\
ta~ic loipa~ic pleura~ic >'isac <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\ 
\kai%
\eix{\gr{ka`i}} t`hn loip`hn gwn'ian\\
t~h| loip~h| gwn'ia|;\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  iki \"u\c cgenin\\
iki a\c c\i s\i\\
iki a\c c\i s\i na e\c sitse,\\
her biri birine,\\
ve bir kenar\\
bir kenara e\c sitse,\\
ya e\c sit a\c c\i lar\i n aras\i nda olan\\
ya da rapteden\\
e\c sit a\c c\i lar\i n birini;\\
kalan kenarlar\i\ da\\
kalan kenarlar\i na e\c sit olacak,\\
%either to either,\\
kalan a\c c\i lar\i\ da\\
kalan a\c c\i lar\i na;\\
\ozqed.
}

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\end{proposition}

\begin{proposition}{27}%Proposition I.27

\newparsen{
>E`an e>ic d'uo e>uje'iac\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t`ac >enall`ax gwn'iac\\
{}>'isac >all'hlaic
poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
par'allhloi >'esontai >all'hlaic\\
a<i e>uje~iai.
}
{
E\u ger iki do\u grunun \"uzerine\\
d\"u\c sen bir do\u gru,\\
ters a\c c\i lar\i\\
birbirine e\c sit
yaparsa,\\
birbirine paralel olacak\\
do\u grular.
}


\newparsen{
E>ic g`ar d'uo e>uje'iac
t`ac AB, GD
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}
<h EZ\\
t`ac >enall`ax gwn'iac
t`ac <up`o AEZ, EZD\\
{}>'isac >all'hlaic
poie'itw;
}
{
\Gar\ iki \gk{AB} ve \gk{GD} do\u grular\i n\i n \"uzerine
d\"u\c sen \gk{EZ},\\
ters
\gk{AEZ} ve \gk{EZD} a\c c\i lar\i n\i\\
birbirine e\c sit
yaps\i n.
}

\newparsen{
l'egw, <'oti\\
par'allhl'oc >estin <h AB t~h| GD.
}
{
\Legohoti\\
\gk{AB}, \gk{GD}'ya paraleldir.
}

\newparsen{
E>i g`ar m'h,\\
{}>ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}\\
a<i AB, GD sumpeso~untai%
\eix{\gr{p'iptw}!\gr{sump'iptw}!\gr{sumpeso~untai} 3rd pl fut ind mid}\\
{}>'htoi >ep`i t`a B, D m'erh\\
{}>`h >ep`i t`a  A, G.\\
{}>ekbebl'hsjwsan%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjwsan} 3rd pl perf imperat mp}\\
\kai%
\eix{\gr{ka`i}} sumpipt'etwsan%
\eix{\gr{p'iptw}!\gr{sump'iptw}!\gr{sumpipt'etwsan} 3rd pl pres imp act}\\
{}>ep`i t`a B, D m'erh
kat`a t`o H.\\
trig'wnou d`h%
\eix{\gr{d'h}} to~u HEZ\\
<h >ekt`oc gwn'ia <h <up`o AEZ
{}>'ish >est`i\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion
t~h| <up`o EZH;\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}};\\
o>uk >'ara\eix{\gr{>'ara}}\\
a<i AB, DG >ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}\\
sumpeso~untai%
\eix{\gr{p'iptw}!\gr{sump'iptw}!\gr{sumpeso~untai} 3rd pl fut ind mid} 
>ep`i t`a B, D m'erh.\\
<omo'iwc d`h%
\eix{\gr{d'h}} deiqj'hsetai%
\eix{\gr{de'iknumi}!\gr{deiqj'hsetai} 3rd sg fut ind
  mp},\verselinebreak <'oti\\
o>ud`e >ep`i t`a A, G;\\
a<i d`e >ep`i mhd'etera t`a m'erh\\
sump'iptousai%
\eix{\gr{p'iptw}!\gr{sump'iptw}!\gr{sump'iptwn} part pres mp}\\
par'allhlo'i e>isin;\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h AB t~h| GD.
}
{
\Gar\ e\u ger de\u gilse,\\
uzat\i lan,\\
\gk{AB} ve \gk{GD} \c carp\i\c sacak,\\
ya \gk B ve \gk D kenar\i nda,\\
ya da \gk A ve \gk G kenar\i nda.\\
Uzat\i lm\i\c s olsun,\\
ve \c carp\i\c s\c s\i n\\
 \gk B ve \gk D taraf\i nda, \gk H'da.\\
 \gk{HEZ} \"u\c cgeninin\\
\gk{AEZ} d\i\c s\ a\c c\i s\i,
e\c sittir\\
i\c c\ ve kar\c s\i t
\gk{EZH}'ya;\\
\imkansiz.\\
\Ara\ \c s\"oyle de\u gildir:\\
\gk{AB} ve \gk{GD}, uzat\i lm\i\c s,\\
\gk B ve \gk D taraf\i nda \c carp\i\c sacak.\\
Benzer \c sekilde \dee\ g\"osterilecek ki\\
\gk A ve \gk G taraf\i nda da de\u gil.\\
Hi\c cbir tarafta\\
\c carp\i\c sanlar,\\
paraleldir;\\
\ara\ \gk{AB}, \gk{GD}'ya paraleldir.\\
}

\newparsen{
>E`an >'ara\eix{\gr{>'ara}} e>ic d'uo e>uje'iac\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t`ac >enall`ax gwn'iac\\
{}>'isac >all'hlaic
poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj},\\
par'allhloi >'esontai\\
a<i e>uje~iai;\\
<'oper >'edei de~ixai.
}
{
E\u ger, \ara,  iki do\u gru \"uzerine\\
d\"u\c sen bir do\u gru\\
ters a\c c\i lar\i\\
birbirine e\c sit yaparsa\\
birbirine paralel olacak\\
do\u grular;\\
\ozqed.
}


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\end{proposition}

\begin{proposition}{28}%Proposition I.28

\newparsen{
>E`an e>ic d'uo e>uje'iac\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t`hn >ekt`oc gwn'ian\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
{}>'ishn poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj}\\
{}>`h t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
dus`in >orja~ic >'isac,\\
par'allhloi >'esontai >all'hlaic\\
a<i e>uje~iai.
}
{
E\u ger iki do\u gru \"uzerine\\
d\"u\c sen bir do\u gru,\\
d\i\c s\ a\c c\i y\i,\\
i\c c\ ve kar\c s\i t\\
ve ayn\i\ tarafta [kalan] a\c c\i ya\\
e\c sit yaparsa,\\
veya i\c c\ ve ayn\i\ tarafta [kalanlar\i]\\
iki dik a\c c\i ya
e\c sit,\\
birbirine paralel olacak\\
do\u grular.
}

\newparsen{
E>ic g`ar d'uo e>uje'iac t`ac AB, GD\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act} <h EZ\\
t`hn >ekt`oc gwn'ian t`hn <up`o EHB\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion gwn'ia|
t~h| <up`o HJD\\
{}>'ishn
poie'itw\\
{}>`h t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
t`ac <up`o BHJ, HJD\\
dus`in >orja~ic >'isac;
}
{
\Gar\  \gk{AB} ve \gk{GD} do\u grular\i\ \"uzerine\\
d\"u\c sen \gk{EZ} do\u grusu,\\
\gk{EHB} d\i\c s\ a\c c\i s\i n\i\\
i\c c\ ve kar\c s\i t
\gk{HJD} a\c c\i s\i na\lli\\
e\c sit yaps\i n,\\
veya i\c c\ ve ayn\i\ tarafta [kalan]\\
\gk{BHJ} ve \gk{HJD} a\c c\i lar\i\\
iki dik a\c c\i ya e\c sit.
}

\newparsen{
l'egw, <'oti\\
par'allhl'oc >estin\\
<h AB t~h| GD.
}
{
\Legohoti\\
paraleldir\\
\gk{AB}, \gk{GD}'ya.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} g`ar >'ish >est`in\\
<h <up`o EHB t~h| <up`o HJD,\\
{}>all`a%
\eix{\gr{>all'a}} <h <up`o EHB\\
t~h| <up`o AHJ >estin >'ish,\\
\kai%
\eix{\gr{ka`i}} <h <up`o AHJ >'ara\eix{\gr{>'ara}}\\
t~h| <up`o HJD >estin >'ish;\\
ka'i e>isin >enall'ax;\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h AB t~h| GD.
}
{
\Gar\ e\c sit oldu\u gundan\\
\gk{EHB}, \gk{HJD}'ya,\\
ama \gk{EHB},\\
\gk{AHJ}'ya e\c sit oldu\u gundan,\\
\ara\ \gk{AHJ} da\\
\gk{HJD}'ya e\c sittir;\\
ve onlar terstir;\\
\ara\ \gk{AB}, \gk{GD}'ya paraleldir.
}

\newparsen{
P'alin, >epe`i%
\eix{\gr{>epe'i}} a<i <up`o BHJ, HJD\\
d'uo >orja~ic >'isai e>is'in,\\
e>is`i d`e \kai%
\eix{\gr{ka`i}} a<i <up`o AHJ, BHJ\\
dus`in >orja~ic >'isai,\\
a<i >'ara\eix{\gr{>'ara}} <up`o AHJ, BHJ\\ 
ta~ic <up`o BHJ, HJD
{}>'isai e>is'in;\\
koin`h\\
{}>afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp}
<h <up`o BHJ;\\
loip`h >'ara\eix{\gr{>'ara}} <h <up`o AHJ\\
loip~h| t~h| <up`o HJD >estin >'ish;\\
ka'i e>isin >enall'ax;\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h AB t~h| GD.
}
{
\Palin\ \gk{BHJ} ve \gk{HJD},\\
iki dik a\c c\i ya e\c sittir,\\
ve  \gk{AHJ} ve \gk{BHJ} de,\\
iki dik a\c c\i ya e\c sittir,\\
\ara\ \gk{AHJ} ve \gk{BHJ},\\
\gk{BHJ} ve \gk{HJD}'ya e\c sittir;\\
ve ortak olarak\\
\gk{BHJ}, ay\i r\i lm\i\c s olsun;\\
\ara\  \gk{AHJ} kalan\i\\
 \gk{HJD} kalan\i na e\c sittir;\\
ve bunlar terstir;\\
\ara\ \gk{AB}, \gk{GD}'ya paraleldir.
}

\newparsen{
>E`an >'ara\eix{\gr{>'ara}} e>ic d'uo e>uje'iac\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t`hn >ekt`oc gwn'ian\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
{}>'ishn poi~h|%
\eix{\gr{poi'ew}!\gr{poi~h"|} 3rd sg pres subj}\\
{}>`h t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
dus`in >orja~ic >'isac,\\
par'allhloi >'esontai\\
a<i e>uje~iai;\\
<'oper >'edei de~ixai.
}
{
E\u ger \ara\ iki do\u gru \"uzerine\\
d\"u\c sen bir do\u gru,\\
d\i\c s\ a\c c\i y\i,\\
i\c c\ ve kar\c s\i t\\
ve ayn\i\ tarafta kalan a\c c\i ya\\
e\c sit yaparsa,\\
veya i\c c\ ve ayn\i\ tarafta kalanlar\i,\\
iki dik a\c c\i ya
e\c sit,\\
birbirine paralel olacak\\
do\u grular;\\
\ozqed.
}
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\end{proposition}

\begin{proposition}{29}%Proposition I.29

\newparsen{
<H e>ic t`ac parall'hlouc e>uje'iac e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t'ac te%
\eix{\gr{te}} >enall`ax gwn'iac\\
{}>'isac >all'hlaic poie~i%
\eix{\gr{poi'ew}}\\
\kai%
\eix{\gr{ka`i}} t`hn >ekt`oc\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion >'ishn\\
\kai%
\eix{\gr{ka`i}} t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
dus`in >orja~ic >'isac.
}
{
Paralel do\u grular \"uzerine d\"u\c sen bir do\u gru\\
hem ters a\c c\i lar\i\\
birbirine e\c sit yapar,\\
hem d\i\c s [a\c c\i]y\i\\
i\c c\ ve kar\c s\i t [a\c c\i]ya e\c sit,\\
hem i\c c\ ve ayn\i\ taraftaki [a\c c\i lar\i]\\
iki dik a\c c\i ya e\c sit.
}


\parsen{
For, on the parallel \strgt s\\
\gk{AB} and \gk{GD}\\
let the \strgt\ \gk{EZ} fall.
}
{
E>ic g`ar parall'hlouc e>uje'iac t`ac AB, GD\\
e>uje~ia >empipt'etw <h EZ;
}
{
\Gar\ paralel \gk{AB} ve \gk{GD} do\u grular\i\ \"uzerine\\
 \gk{EZ} do\u grusu d\"u\c ss\"un.
}

\parsen{
I say that\\
the alternate angles\\
\gk{AHJ} and \gk{HJD}\\
equal\\
it makes,\\
and the exterior angle \gk{EHB}\\
to the interior and opposite \gk{HJD}\\
equal,\\
and the interior and in the same parts\\
\gk{BHJ} and \gk{HJD}\\
to two \rgt s equal.
}
{
l'egw, <'oti
t`ac >enall`ax gwn'iac t`ac <up`o AHJ, HJD >'isac poie~i%
\eix{\gr{poi'ew}}\\
\kai%
\eix{\gr{ka`i}} t`hn >ekt`oc gwn'ian t`hn <up`o EHB\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
t~h| <up`o HJD >'ishn\\
\kai%
\eix{\gr{ka`i}} t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
t`ac <up`o BHJ, HJD\\
dus`in >orja~ic >'isac. 
}
{
\Legohoti\
ters \gk{AHJ} ve \gk{HJD} a\c c\i lar\i\ e\c sit yapar,\\
ve \gk{EHB} d\i\c s\ a\c c\i s\i n\i\\
i\c c\ ve kar\c s\i t\\
\gk{HJD}'ya e\c sit,\\
ve i\c c\ ve ayn\i\ taraftaki\\
\gk{BHJ} ile \gk{HJD} a\c c\i lar\i n\i\\
iki dik a\c c\i ya e\c sit.
}

\parsen{
For, if it is unequal,\\
\gk{AHJ} to \gk{HJD},\\
one of them is greater.\\
Let the greater be \gk{AHJ};\\
let be added in common\\
\gk{BHJ};\\
therefore \gk{AHJ} and \gk{BHJ}\\
than \gk{BHJ} and \gk{HJD}\\
are greater.\\
However, \gk{AHJ} and \gk{BHJ}\\
to two \rgt s\\
equal are.\\
Therefore [also] \gk{BHJ} and \gk{HJD}\\
than two \rgt s\\
less are.\\
And [\strgt s] from [angles] that are less\\
than two \rgt s,\\
extended to the infinite,\\
fall together.\\
Therefore \gk{AB} and \gk{GD},\\
extended to the infinite,\\
will fall together.\\
But they do not fall together,\\
by their being assumed parallel.\\
Therefore is not unequal\\
\gk{AHJ} to \gk{HJD}.\\
Therefore it is equal.\\
However, \gk{AHJ} to \gk{EHB}\\
is equal;\\
therefore also \gk{EHB} to \gk{HJD}\\
is equal;\\
let \gk{BHJ} be added in common;\\
therefore \gk{EHB} and \gk{BHJ}\\
to \gk{BHJ} and \gk{HJD}\\
is equal.\\
But \gk{EHB} and \gk{BHJ}\\
to two \rgt s\\
are equal.\\
Therefore also \gk{BHJ} and \gk{HJD}\\
to two \rgt s\\
are equal.
}
{
E>i g`ar >'anis'oc >estin\\
<h <up`o AHJ t~h| <up`o HJD,\\
m'ia a>ut~wn me'izwn >est'in.\\
{}>'estw me'izwn <h <up`o AHJ;\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres
  imperat mp} <h <up`o BHJ;\\
a<i >'ara\eix{\gr{>'ara}} <up`o AHJ, BHJ\\
t~wn <up`o BHJ, HJD
me'izon'ec e>isin.\\
{}>all`a%
\eix{\gr{>all'a}} a<i <up`o AHJ, BHJ\\
dus`in >orja~ic >'isai e>is'in.\\
{}[\kai%
\eix{\gr{ka`i}}] a<i >'ara\eix{\gr{>'ara}} <up`o BHJ, HJD\\
d'uo >orj~wn >el'asson'ec e>isin.\\
a<i d`e >ap> >elass'onwn >`h d'uo >orj~wn\\
{}>ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}
e>ic >'apeiron\\
sump'iptousin;\\
a<i >'ara\eix{\gr{>'ara}} AB, GD\\
{}>ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp} 
e>ic >'apeiron\\
sumpeso~untai;\\
o>u sump'iptousi d`e\\
di`a\eix{\gr{di'a}} t`o parall'hlouc a>ut`ac\\
<upoke~isjai%
\eix{\gr{ke~imai}!\gr{<upoke~imai}!\gr{<upoke~isjai} pres inf mp};\\
o>uk >'ara\eix{\gr{>'ara}} >'anis'oc >estin\\
<h <up`o AHJ t~h| <up`o HJD;\\
{}>'ish >'ara\eix{\gr{>'ara}}.\\
{}>all`a%
\eix{\gr{>all'a}} <h <up`o AHJ\\
t~h| <up`o EHB >estin >'ish;\\
\kai%
\eix{\gr{ka`i}} <h <up`o EHB >'ara\eix{\gr{>'ara}}\\
t~h| <up`o HJD >estin >'ish;\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o BHJ;\\
a<i >'ara\eix{\gr{>'ara}} <up`o EHB, BHJ\\
ta~ic <up`o BHJ, HJD >'isai e>is'in.\\
{}>all`a%
\eix{\gr{>all'a}} a<i <up`o EHB, BHJ\\
d'uo >orja~ic >'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o BHJ, HJD >'ara\eix{\gr{>'ara}}\\
d'uo >orja~ic >'isai e>is'in.
}
{
\Gar\ e\u ger e\c sit de\u gilse\\
\gk{AHJ}, \gk{HJD} a\c c\i s\i na,\\
biri b\"uy\"ukt\"ur.\\
\gk{AHJ} daha b\"uy\"uk olsun;\\
ortak olarak \gk{BHJ} eklenmi\c s\ olsun;\\
\ara\ \gk{AHJ} ve \gk{BHJ},\\
 \gk{BHJ} ve \gk{HJD}'dan b\"uy\"ukt\"ur.\\
Ama \gk{AHJ} ve \gk{BHJ}\\
iki dik a\c c\i ya e\c sittir.\\
\Ara\  \gk{BHJ} ve \gk{HJD} [da]\\
iki dik a\c c\i dan k\"u\c c\"ukt\"ur.\\
Ve iki dik a\c c\i dan k\"u\c c\"uk [a\c c\i lar]dan\\
sonsuza uzat\i lan [do\u grular],\\
\c carp\i\c s\i r.\\
\Ara\ \gk{AB} ve \gk{GD},\\
uzat\i l\i nca sonsuza,\\
\c carp\i\c s\i r.\\
Ama \c carp\i\c smaz,\\
\dia\ paralel\\
kabul edilir.\\
\Ara\ e\c sit de\u gil de\u gildir\\
\gk{AHJ}, \gk{HJD}'ya.\\
\Ara\ e\c sittir.\\
Ama \gk{AHJ},\\
\gk{EHB} a\c c\i s\i na e\c sittir;\\
\ara\ \gk{EHB} da\\
\gk{HJD} a\c c\i s\i na e\c sittir;\\
ortak olarak \gk{BHJ} eklenmi\c s\ olsun;\\
\ara\ \gk{EHB} ve \gk{BHJ},\\
 \gk{BHJ} ve \gk{HJD}'ya e\c sittir.\\
Ama \gk{EHB} ve \gk{BHJ}\\
iki dik a\c c\i ya e\c sittir.\\
\Ara\  \gk{BHJ} ve \gk{HJD} da\\
iki dik a\c c\i ya e\c sittir.
}

\parsen{
Therefore the on-parallel-\strgt s \strgt\\
falling\\
the alternate angles\\
makes equal to one another,\\
and the exterior\\
to the interior and opposite\\
equal,\\
and the interior and in the same parts\\
to two \rgt s equal;\\
\myqed
}
{
<H  >'ara\eix{\gr{>'ara}} e>ic t`ac parall'hlouc e>uje'iac e>uje~ia
>emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act}\\
t'ac te%
\eix{\gr{te}} >enall`ax gwn'iac\\
{}>'isac >all'hlaic poie~i%
\eix{\gr{poi'ew}}\\
\kai%
\eix{\gr{ka`i}} t`hn >ekt`oc\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion >'ishn\\
\kai%
\eix{\gr{ka`i}} t`ac >ent`oc \kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
dus`in >orja~ic >'isac;\\
<'oper >'edei de~ixai.
}
{
\Ara\ paralel do\u grular \"uzerine d\"u\c sen bir do\u gru\\
hem ters a\c c\i lar\i\\
birbirine e\c sit yapar,\\
hem  d\i\c s [a\c c\i]y\i\\
i\c c\ ve kar\c s\i t [a\c c\i]ya e\c sit,\\
hem i\c c\ ve ayn\i\ taraftaki [a\c c\i lar\i]\\
iki dik a\c c\i ya e\c sit;\\
\ozqed.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(0.11828125,-0.35421875){\gk G}
\usefont{T1}{ptm}{m}{n}
\rput(4.9371877,-0.39421874){\gk D}
\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(1.7403125,0.8857812){\gk H}
\usefont{T1}{ptm}{m}{n}
\rput(3.1825,-0.13421875){\gk J}
\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{30}%Proposition I.30

\newparsen{
A<i t~h| a>ut~h| e>uje'ia| par'allhloi\\
\kai%
\eix{\gr{ka`i}} >all'hlaic e>is`i par'allh\-loi.
}
{
Ayn\i\ do\u gruya paraleller,\\
birbirine de paraleldir.
}

\newparsen{
>'Estw\\
<ekat'era t~wn AB, GD\\
t~h| EZ par'allhloc;
}
{
Olsun\\
\gk{AB} ve \gk{GD}'n\i n her biri,\\
\gk{EZ}'ya paralel.
}

\newparsen{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} <h AB t~h| GD >esti par'allhloc.
}
{
\Legohoti\\
\gk{AB} da \gk{GD}'ya paraleldir.
}


\newparsen{
>Empipt'etw g`ar\\
e>ic a>ut`ac e>uje~ia <h HK.
}
{
\Gar\ d\"u\c ss\"un\\
\"uzerlerine \gk{HK}.
}

\parsen{
Then, since on the parallel \strgt s\\
\gk{AB} and \gk{EZ}\\
a \strgt\ has fallen, [namely] \gk{HK},\\
equal therefore is \gk{AHK} to \gk{HJZ}.\\
Moreover,\\
since on the parallel \strgt s\\
\gk{EZ} and \gk{GD}\\
a \strgt\ has fallen, [namely] \gk{HK},\\
equal is \gk{HJZ} to \gk{HKD}.\\
And it was shown also that\\
\gk{AHK} to \gk{HJZ} is equal.\\
Also \gk{AHK} therefore to \gk{HKD}\\
is equal;\\
and they are alternate.\\
Parallel therefore is \gk{AB} to \gk{GD}.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} e>ic parall'hlouc e>uje'iac
t`ac AB, EZ\\
e>uje~ia >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act} <h HK,\\
{}>'ish >'ara\eix{\gr{>'ara}} <h <up`o AHK t~h| <up`o HJZ.\\
p'alin\eix{\gr{p'alin}},
{}>epe`i%
\eix{\gr{>epe'i}} e>ic parall'hlouc e>uje'iac
t`ac EZ, GD\\
e>uje~ia >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act} <h HK,\\
{}>'ish >est`in <h <up`o HJZ t~h| <up`o HKD.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}}\\
<h <up`o AHK t~h| <up`o HJZ >'ish.\\
\kai%
\eix{\gr{ka`i}} <h <up`o AHK >'ara\eix{\gr{>'ara}}\\
t~h| <up`o HKD >estin >'ish;\\
ka'i e>isin >enall'ax.\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h AB t~h| GD.
}
{
Ve paralel
\gk{AB} ve \gk{EZ} do\u grular\i n\i n \"uzerine\\
\gk{HK} do\u grusu d\"u\c sm\"u\c s\ oldu\u gundan,\\
\ara\ \gk{AHK}, \gk{HJZ}'ya e\c sittir.\\
\Palin,
paralel \gk{EZ} ve \gk{GD} do\u grular\i n\i n \"uzerine\\
\gk{HK} do\u grusu d\"u\c sm\"u\c s\ oldu\u gundan,\\
\gk{HJZ}, \gk{HKD} a\c c\i s\i na e\c sittir.\\
Ve g\"osterilmi\c sti\\
\gk{AHK}, \gk{HJZ}'ya e\c sit.\\
Ve \ara\ \gk{AHK},\\
\gk{HKD}'ya e\c sittir;\\
ve bunlar terstir.\\
\Ara\ \gk{AB}, \gk{GD}'ya paraleldir.
}

\parsen{
Therefore [\strgt s]\\
to the same \strgt\\
parallel\\
also to one another are parallel;\\
\myqed
}
{
[A<i  >'ara\eix{\gr{>'ara}}
t~h| a>ut~h| e>uje'ia| par'allhloi\\
\kai%
\eix{\gr{ka`i}} >all'hlaic e>is`i par'allhloi;]\\
<'oper >'edei de~ixai.
}
{
\Ara\
ayn\i\ do\u gruya paraleller\\
birbirine de paraleldir;\\
\ozqed.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
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\usefont{T1}{ptm}{m}{n}
\rput(0.10828125,-0.975){\gk G}
\usefont{T1}{ptm}{m}{n}
\rput(5.0271873,-0.915){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(0.18921874,-0.155){\gk E}
\usefont{T1}{ptm}{m}{n}
\rput(5.049844,-0.135){\gk Z}
\usefont{T1}{ptm}{m}{n}
\rput(2.9103124,1.085){\gk H}
\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(2.0157812,-0.735){\gk K}
\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{31}%Proposition I.31

\newparsen{
Di`a to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme'iou\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| par'allhlon\\
e>uje~ian gramm`hn >agage~in\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
Verilmi\c s bir noktadan\\
verilmi\c s bir do\u gruya paralel\\
bir do\u gru \c cizgi ilerlemek.
}


\newparsen{
>'Estw\\
t`o m`en%
\eix{\gr{m'en}} doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} shme~ion t`o A,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h BG;
}
{
Olsun\\
verilmi\c s nokta \gk A,\\
ve verilmi\c s do\u gru \gk{BG}.
}

\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}} \\
di`a to~u A shme'iou\\
t~h| BG e>uje'ia| par'allhlon\\
e>uje~ian gramm`hn >agage~in\eix{\gr{>'agw}!\gr{>agage~in} aor inf act}.
}
{
\Deidee\\
\gk A noktas\i ndan\\
 \gk{BG} do\u grusuna paralel\\
bir do\u gru \c cizgi ilerlemek.
}

\parsen{
Suppose there has been chosen\\
on \gk{BG}\\
a random point \gk D,\\
and there has been joined \gk{AD}.\\
and there has been constructed,\\
on the \strgt\ \gk{DA},\\
and at the point \gk A of it,\\
to the angle \gk{ADG} equal,\\
\gk{DAE};\\
and suppose there has been extended,\\
in \strgt s with \gk{EA},\\
the \strgt\ \gk{AZ}.
}
{
E>il'hfjw%
\eix{\gr{lamb'anw}!\gr{e>il'hfjw} 3rd sg perf imperat mp}\\
{}>ep`i t~hc BG\\
tuq`on shme~ion t`o D,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h AD;\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
pr`oc t~h| DA e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| A\\
t~h| <up`o ADG gwn'ia| >'ish\\
<h <up`o DAE;\\
\kai%
\eix{\gr{ka`i}} >ekbebl'hsjw%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjw} 3rd sg perf imperat mp}\\
{}>ep> e>uje'iac t~h| EA\\
e>uje~ia <h AZ.
}
{
al\i nm\i\c s olsun\\
 \gk{BG} \"uzerinde\\
rastgele bir  \gk D noktas\i,\\
ve \gk{AD} birle\c stirilmi\c s\ olsun,\\
ve in\c sa edilmi\c s\ olsun,\\
\gk{DA} do\u grusunda,\\
ve onun \gk A noktas\i nda,\\
\gk{ADG} a\c c\i s\i na e\c sit,\\
\gk{DAE};\\
ve uzat\i lm\i\c s\ olsun,\\
\gk{EA} ile ayn\i\ do\u gruda,\\
\gk{AZ} do\u grusu.
}

\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}}
e>ic d'uo e>uje'iac t`ac BG, EZ\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act} <h AD\\
t`ac >enall`ax gwn'iac
t`ac <up`o EAD, ADG\\
{}>'isac >all'hlaic pepo'ihken,\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h EAZ t~h| BG.
}
{
Ve \gk{BG} ve \gk{EZ} do\u grular\i\ \"uzerine\\
d\"u\c sen \gk{AD} do\u grusu,\\
ters \gk{EAD} ve \gk{ADG} a\c c\i lar\i n\i\\
birbirine e\c sit yapt\i\u g\i ndan,\\
\ara\ \gk{EAZ}, \gk{BG}'ya paraleldir.
}

\newparsen{
Di`a to~u doj'entoc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} >'ara\eix{\gr{>'ara}} shme'iou to~u A\\
t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'ia| t~h| BG par'allhloc\\
e>uje~ia gramm`h >~hktai%
\eix{\gr{>'agw}!\gr{>~hktai} 3rd sg perf ind mp} <h EAZ;\\
<'oper >'edei poi~hsai.
}
{
\Ara,  verilmi\c s \gk A noktas\i ndan,\\
verilmi\c s \gk{BG} do\u grusuna paralel,\\
do\u gru \gk{EAZ} \c cizgisi, ilerletilmi\c s\ oldu;\\
\ozqef.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-2.401875)(8.692187,2.401875)
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\usefont{T1}{ptm}{m}{n}
\rput(3.080625,2.223125){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(0.10125,0.463125){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(4.8951564,0.483125){\gk G}
\usefont{T1}{ptm}{m}{n}
\rput(2.1540625,0.283125){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(0.13609375,2.003125){\gk E}
\usefont{T1}{ptm}{m}{n}
\rput(4.996719,2.003125){\gk Z}
\psline[linewidth=0.04cm](2.9740624,1.998125)(2.2340624,0.498125)
\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{32}%Proposition I.32

\newparsen{
Pant`oc trig'wnou\\
mi~ac t~wn pleur~wn prosekblhje'ishc%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass}\\
<h >ekt`oc gwn'ia\\
dus`i ta~ic >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
{}>'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} a<i >ent`oc to~u trig'wnou tre~ic gwn'iai\\
dus`in >orja~ic >'isai e>is'in.
}
{
Herhangi bir \"u\c cgenin\\
kenarlar\i ndan biri uzat\i l\i nca,\\
d\i\c s\ a\c c\i\\
iki kar\c s\i t i\c c\ a\c c\i ya\\
e\c sittir,\\
ve \"u\c cgenin \"u\c c\ i\c c\ a\c c\i s\i\lli\\
iki dik a\c c\i ya e\c sittir.
}

\newparsen{
>'Estw\\
tr'igwnon t`o ABG,\\
\kai%
\eix{\gr{ka`i}} prosekbebl'hsjw%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekbebl'hsjw} 3rd sg perf imperat mp}\\
a>uto~u m'ia pleur`a <h BG >ep`i t`o D;
}
{
Olsun\\
\"u\c cgen \gk{ABG},\\
ve uzat\i lm\i\c s\ olsun\\
onun \gk{BG} kenar\i, \gk D noktas\i na.\\
}

\newparsen{
l'egw, <'oti\\
<h >ekt`oc gwn'ia <h <up`o AGD >'ish >est`i\\
dus`i ta~ic >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
ta~ic <up`o GAB, ABG,\\
\kai%
\eix{\gr{ka`i}} a<i >ent`oc to~u trig'wnou tre~ic gwn'iai\\
a<i <up`o ABG, BGA, GAB\\
dus`in >orja~ic >'isai e>is'in.
}
{
\Legohoti\\
 \gk{AGD} d\i\c s\ a\c c\i s\i\ e\c sittir\\
iki i\c c\ ve kar\c s\i t\\
\gk{GAB} ve \gk{ABG} a\c c\i lar\i na,\\
ve \"u\c cgenin \"u\c c\ i\c c\ a\c c\i s\i\lli\\
---\gk{ABG}, \gk{BGA}, ve \gk{GAB}---,\\
iki dik a\c c\i ya e\c sittir.
}

\newparsen{
>'Hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} g`ar\\
di`a\eix{\gr{di'a}} to~u G shme'iou\\
t~h| AB e>uje'ia| par'allhloc\\
<h GE.
}
{
\Gar\ ilerletilmi\c s olsun\\
 \gk G noktas\i ndan\\
 \gk{AB} do\u grusuna paralel\\
\gk{GE}.
}

\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} par'allhl'oc >estin\\
<h AB t~h| GE,\\
\kai%
\eix{\gr{ka`i}} e>ic a>ut`ac >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act}\\
<h AG,\\
a<i >enall`ax gwn'iai a<i <up`o BAG, AGE\\
{}>'isai >all'hlaic e>is'in.\\
p'alin\eix{\gr{p'alin}}, >epe`i%
\eix{\gr{>epe'i}} par'allhl'oc >estin\\
<h AB t~h| GE,\\
\kai%
\eix{\gr{ka`i}} e>ic a>ut`ac >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act}\\
e>uje~ia <h BD,\\
<h >ekt`oc gwn'ia <h <up`o EGD >'ish >est`i\\
t~h| >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion t~h| <up`o ABG.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}}\\
<h <up`o AGE t~h| <up`o BAG >'ish;\\
<'olh >'ara\eix{\gr{>'ara}} <h <up`o AGD gwn'ia\\
{}>'ish >est`i\\
dus`i ta~ic >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
ta~ic <up`o BAG, ABG.
}
{
Ve paralel oldu\u gundan\\
\gk{AB}, \gk{GE}'a,\\
ve bunlar\i n \"uzerine d\"u\c st\"u\u g\"unden\\
\gk{AG},\\
ters \gk{BAG} ve \gk{AGE} a\c c\i lar\i\\
birbirine e\c sittir.\\
\Palin, paralel oldu\u gundan\\
\gk{AB}, \gk{GE} do\u grusuna,\\
ve bunlar\i n \"uzerine d\"u\c st\"u\u g\"unden\\
 \gk{BD} do\u grusu,\\
\gk{EGD} d\i\c s\ a\c c\i s\i\ e\c sittir\\
i\c c\ ve kar\c s\i t \gk{ABG} a\c c\i s\i na.\\
Ve g\"osterilmi\c sti\\
\gk{AGE} da, \gk{BAG} a\c c\i s\i na e\c sit.\\
\Ara\ b\"ut\"un \gk{AGD} a\c c\i s\i\\
e\c sittir\\
iki i\c c\ ve kar\c s\i t\\
\gk{BAG} ve \gk{ABG} a\c c\i lar\i na.
}

\newparsen{
Koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o AGB;\\
a<i >'ara\eix{\gr{>'ara}} <up`o AGD, AGB\\
tris`i ta~ic <up`o ABG, BGA, GAB\\
{}>'isai e>is'in.\\
{}>all>%
\eix{\gr{>all'a}} a<i <up`o AGD, AGB\\
dus`in >orja~ic {}>'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o AGB, GBA, GAB >'ara\eix{\gr{>'ara}}\\
dus`in >orja~ic >'isai e>is'in.
}
{
Ortak olarak \gk{AGB} eklensin;\\
\ara\ \gk{AGD} ve \gk{AGB} a\c c\i lar\i\\
 \gk{ABG}, \gk{BGA} ve \gk{GAB} \"u\c cl\"us\"une\\
e\c sittir.\\
Ama \gk{AGD} ve \gk{AGB},\\
iki dik a\c c\i ya e\c sittir;\\
\ara\ \gk{AGB}, \gk{GBA} ve \gk{GAB} da\\
iki dik a\c c\i ya e\c sittir.
}

\newparsen{
Pant`oc >'ara\eix{\gr{>'ara}} trig'wnou\\
mi~ac t~wn pleur~wn prosekblhje'ishc%
\eix{\gr{b'allw}!\gr{prosekb'allw}!\gr{prosekblhje'ic} part aor pass}\\
<h >ekt`oc gwn'ia\\
dus`i ta~ic >ent`oc \kai%
\eix{\gr{ka`i}} >apenant'ion\\
{}>'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} a<i >ent`oc to~u trig'wnou tre~ic gwn'iai\\
dus`in >orja~ic >'isai e>is'in;\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir \"u\c cgenin\\
kenarlar\i ndan biri uzat\i l\i nca,\\
d\i\c s\ a\c c\i\\
iki kar\c s\i t i\c c\ a\c c\i ya\\
e\c sittir,\\
ve \"u\c cgenin \"u\c c\ i\c c\ a\c c\i s\i\\
iki dik a\c c\i ya e\c sittir;\\
\ozqed.
}
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\end{proposition}

\begin{proposition}{33}%Proposition I.33

\newparsen{
A<i t`ac >'isac te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} parall'hlouc\\
{}>ep`i t`a a>ut`a m'erh >epizeugn'uousai%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epizeugn'uwn} part pres act}\\
e>uje~iai \kai%
\eix{\gr{ka`i}} a>uta`i\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhlo'i e>isin.
}
{
E\c sit paralelleri\\
ayn\i\ tarafta birle\c stiren\\
do\u grular\i n kendileri de\\
hem e\c sit hem paraleldirler.
}

\parsen{
Let be\\
equals and parallels\\
\gk{AB} and \gk{GD},\\
and let join these\\
in the same parts\\
\strgt s \gk{AG} and \gk{BD}.
}
{
>'Estwsan\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhloi\\
a<i  AB, GD,\\
\kai%
\eix{\gr{ka`i}} >epizeugn'utwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epizeugn'utwsan} 3rd pl pres imperat mp} a>ut`ac\\
{}>ep`i t`a a>ut`a m'erh\\
e>uje~iai a<i AG, BD;
}
{
Olsun\\
e\c sit paraleller\\
\gk{AB} ve \gk{GD},\\
ve bunlar\i\ birle\c stirsin\\
ayn\i\ tarafta\\
\gk{AG} ve \gk{BD} do\u grular\i.
}

\parsen{
I say that\\
also \gk{AG} and \gk{BD}\\
equal and parallel are.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} a<i AG, BD\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhlo'i e>isin.
}
{
\Legohoti\\
 \gk{AG} ve \gk{BD} da\\
e\c sit ve paraleldir.
}


\parsen{
Suppose there has been joined \gk{BG}.\\
And since parallel is \gk{AB} to \gk{GD},\\
and on these has fallen \gk{BG},\\
the alternate angles \gk{ABG} and \gk{BGD}\\
equal to one another are.\\
And since equal is \gk{AB} to \gk{GD},\\
and common [is] \gk{BG},\\
then the two \gk{AB} and \gk{BG}\\
to the two \gk{BG} and \gk{GD}\\
equal are;\\
also angle \gk{ABG}\\
to angle \gk{BGD}\\
{}[is] equal;\\
therefore the base \gk{AG}\\
to the base \gk{BD}\\
is equal,\\
and the triangle \gk{ABG}\\
to the triangle \gk{BGD}\\
is equal,\\
and the remaining angles\\
to the remaining angles\\
equal will be,\\
either to either,\\
which the equal sides subtend;\\
equal therefore\\
the \gk{AGB} angle to \gk{GBD}.\\
And since on the two \strgt s\\
\gk{AG} and \gk{BD}\\
the \strgt\ falling---\gk{BG}---\\
alternate angles equal to one another\\
has made,\\
parallel therefore is \gk{AG} to \gk{BD}.\\
And it was shown to it also equal.
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h BG.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} par'allhl'oc >estin\\
<h AB t~h| GD,\\
\kai%
\eix{\gr{ka`i}} e>ic a>ut`ac >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act}\\
<h BG,\\
a<i >enall`ax gwn'iai a<i <up`o ABG, BGD\\
{}>'isai >all'hlaic e>is'in.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in <h AB t~h| GD\\
koin`h d`e <h BG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i AB, BG\\
d'uo ta~ic BG, GD >'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ABG\\
gwn'ia| t~h| <up`o BGD >'ish;\\
b'asic >'ara\eix{\gr{>'ara}} <h AG\\
b'asei t~h| BD >estin >'ish,\\
\kai%
\eix{\gr{ka`i}} t`o ABG tr'igwnon\\
t~w| BGD trig'wnw| >'ison >est'in,\\
\kai%
\eix{\gr{ka`i}} a<i loipa`i gwn'iai\\
ta~ic loipa~ic gwn'iaic >'isai >'esontai\\
<ekat'era <ekat'era|,\\
<uf> <`ac a<i >'isai pleura`i <upote'inousin%
\eix{\gr{te'inw}!\gr{<upote'inw}};\\
{}>'ish >'ara\eix{\gr{>'ara}}\\
<h <up`o AGB gwn'ia t~h| <up`o GBD.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} e>ic d'uo e>uje'iac t`ac AG, BD\\
e>uje~ia >emp'iptousa%
\eix{\gr{p\i ptw}!\gr{>emp'iptw}!\gr{>emp'iptwn} part pres act} <h BG\\
t`ac >enall`ax gwn'iac >'isac >all'hlaic\\
pepo'ihken,\\
par'allhloc >'ara\eix{\gr{>'ara}} >est`in <h AG t~h| BD.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e a>ut~h| \kai%
\eix{\gr{ka`i}} >'ish.
}
{
\gk{BG} birle\c stirilmi\c s olsun.\\
Ve paralel oldu\u gundan\\
\gk{AB}, \gk{GD}'ya,\\
ve bunlar\i n \"uzerine d\"u\c st\"u\u g\"unden\\
\gk{BG},\\
ters \gk{ABG} ve \gk{BGD} a\c c\i lar\i\\
birbirine e\c sittir.\\
Ve \gk{AB}, \gk{GD}'ya e\c sit oldu\u gundan,\\
ve \gk{BG} ortak [oldu\u gundan],\\
\gk{AB} ve \gk{BG} ikilisi\\
\gk{BG} ve \gk{GD} ikilisine e\c sittir;\\
\gk{ABG} a\c c\i s\i\ da\\
 \gk{BGD} a\c c\i s\i na e\c sittir;\\
\ara\ \gk{AG} taban\i\\
 \gk{BD} taban\i na e\c sittir,\\
ve \gk{ABG} \"u\c cgeni\\
 \gk{BGD} \"u\c cgenine e\c sittir,\\
ve kalan a\c c\i lar\\
kalan a\c c\i lara
e\c sit olacak,\\
her biri birine,\\
e\c sit kenarlar\i n raptetti\u gi;\\
\ara\ e\c sittir\\
\gk{AGB} a\c c\i s\i, \gk{GBD}'ya.\\
Ve iki \gk{AG} ve \gk{BD} do\u grular\i n\i n \"uzerine\\
d\"u\c sen \gk{BG} do\u grusu,\\
ters a\c c\i lar\i\ birbirine e\c sit\\
yapt\i\u g\i ndan,\\
\ara\ \gk{AG}, \gk{BD}'ya paraleldir.\\
Ve ona e\c sit oldu\u gu da g\"osterilmi\c sti.
}

\parsen{
Therefore \strgt s joining equals and parallels to the same parts\\
also themselves equal and parallel are.
\myqed
}
{
A<i >'ara\eix{\gr{>'ara}} t`ac >'isac te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} parall'hlouc\\
{}>ep`i t`a a>ut`a m'erh >epizeugn'uousai%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epizeugn'uwn} part pres act}\\
e>uje~iai \kai%
\eix{\gr{ka`i}} a>uta`i\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhlo'i e>isin;\\
<'oper >'edei de~ixai.
}
{
\Ara\ e\c sit paralelleri\\
ayn\i\ tarafta birle\c stiren\\
do\u grular\i n kendileri de\\
hem e\c sit hem paraleldirler;\\
\ozqed.
}
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\end{proposition}

\begin{proposition}{34}%Proposition I.34

\newparsen{
T~wn parallhlogr'ammwn qwr'iwn\\
a<i >apenant'ion pleura'i te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} gwn'iai\\
{}>'isai >all'hlaic e>is'in,\\
\kai%
\eix{\gr{ka`i}} <h di'ametroc a>ut`a d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}.
}
{
Paralelkenar alanlar\i n\\
hem kar\c s\i t kenarlar\i\ hem de a\c c\i lar\i,\\
birbirine e\c sittir,\\
ve k\"o\c segen onlar\i\ ikiye b\"oler.
}

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\end{center}



\newparsen{
>'Estw\\
parallhl'ogrammon qwr'ion
t`o AGDB,\\
di'amet\-roc d`e a>uto~u <h BG;
}
{
Olsun\\
paralelkenar alan \gk{AGDB};\\
ve onun k\"o\c segeni, \gk{BG}.
}

\newparsen{
l'egw, <'oti\\
to~u AGDB parallhlogr'ammou\\
a<i >apenant'ion pleura'i te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} gwn'iai\\
{}>'isai >all'hlaic e>is'in,\\
\kai%
\eix{\gr{ka`i}} <h BG di'ametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}.
}
{
\Legohoti\\
 \gk{AGDB} paralelkenar\i n\i n\\
kar\c s\i t kenarlar\i\ ve a\c c\i lar\i\\
birbirine e\c sittir,\\
ve \gk{BG} k\"o\c segeni onu ikiye b\"oler.
}

\parsen{
For, since parallel is\\
\gk{AB} to \gk{GD},\\
and on these has fallen\\
a \strgt, \gk{BG},\\
the alternate angles \gk{ABG} and \gk{BGD}\\
equal to one another are.\\
Moreover, since parallel is\\
\gk{AG} to \gk{BD},\\
and on these has fallen\\
\gk{BG},\\
the alternate angles \gk{AGB} and \gk{GBD}\\
equal to one another are.\\
Then two triangles there are,\\
\gk{ABG} and \gk{BGD},\\
the two angles \gk{ABG} and \gk{BGA}\\
to the two \gk{BGD} and \gk{GBD}\\
equal having,\\
either to either,\\
and one side to one side equal,\\
that near the equal angles,\\
their common \gk{BG};\\
also then the remaining sides\\
to the remaining sides\\
equal they will have,\\
either to either,\\
and the remaining angle\\
to the remaining angle;\\
equal, therefore,\\
the \gk{AB} side to \gk{GD},\\
and \gk{AG} to \gk{BD},\\
and yet equal is the \gk{BAG} angle\\
to \gk{GDB}.\\
And since equal is the \gk{ABG} angle\\
to \gk{BGD},\\
and \gk{GBD} to \gk{AGB},\\
therefore the whole \gk{ABD}\\
to the whole \gk{AGD}\\
is equal.\\
And was shown also\\
\gk{BAG} to \gk{GDB} equal.
}
{
>Epe`i%
\eix{\gr{>epe'i}} g`ar par'allhl'oc >estin\\
<h AB t~h| GD,\\
\kai%
\eix{\gr{ka`i}} e>ic a>ut`ac >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act}
e>uje~ia <h BG,\lli\\
a<i >enall`ax gwn'iai a<i <up`o ABG, BGD\\
{}>'isai >all'hlaic e>is'in.\\
p'alin\eix{\gr{p'alin}} >epe`i%
\eix{\gr{>epe'i}} par'allhl'oc >estin\\
<h AG t~h| BD,\\
\kai%
\eix{\gr{ka`i}} e>ic a>ut`ac >emp'eptwken%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>emp'eptwken} 3rd sg perf ind act}
<h BG,\lli\\
a<i >enall`ax gwn'iai a<i <up`o AGB, GBD\\
{}>'isai >all'hlaic e>is'in.\\
d'uo d`h%
\eix{\gr{d'h}} tr'igwn'a >esti\\
t`a ABG, BGD\\
t`ac d'uo gwn'iac t`ac <up`o ABG, BGA\\
dus`i ta~ic <up`o BGD, GBD\\
{}>'isac >'eqonta%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\\
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} m'ian pleur`an mi~a| pleur~a| >'ishn\\
t`hn pr`oc ta~ic >'isaic gwn'iaic\\
koin`hn a>ut~wn t`hn BG;\\
\kai%
\eix{\gr{ka`i}} t`ac loip`ac >'ara\eix{\gr{>'ara}} pleur`ac\\
ta~ic loipa~ic >'isac <'exei%
\eix{\gr{>'eqw}!\gr{>'exei} 3rd sg fut ind act}\\
<ekat'eran <ekat'era|\\
\kai%
\eix{\gr{ka`i}} t`hn loip`hn gwn'ian\\
t~h| loip~h| gwn'ia|;\\
{}>'ish >'ara\eix{\gr{>'ara}}\\
<h m`en%
\eix{\gr{m'en}} AB pleur`a t~h| GD,\\
<h d`e AG t~h| BD,\\
\kai%
\eix{\gr{ka`i}} >'eti%
\eix{\gr{>'eti}} 
>'ish >est`in\\
<h <up`o BAG gwn'ia t~h| <up`o GDB.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} <up`o ABG gwn'ia t~h| <up`o BGD,\\
<h d`e <up`o GBD t~h| <up`o AGB,\\
<'olh >'ara\eix{\gr{>'ara}} <h <up`o ABD\\
<'olh| t~h| <up`o AGD
{}>estin >'ish.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}}\\
<h <up`o BAG t~h| <up`o GDB >'ish.
}
{
\Gar\ paralel oldu\u gundan\\
\gk{AB}, \gk{GD}'ya,\\
ve bunlar\i n \"uzerine d\"u\c sm\"u\c s oldu\u gundan \gk{BG},\\
ters \gk{ABG} ve \gk{BGD} a\c c\i lar\i\\
birbirine e\c sittir.\\
\Palin, paralel oldu\u gundan\\
\gk{AG}, \gk{BD}'ya,\\
ve bunlar\i n \"uzerine d\"u\c sm\"u\c s oldu\u gundan \gk{BG},\\
ters \gk{AGB} ve \gk{GBD} a\c c\i lar\i\\
birbirine e\c sittir.\\
\Dee\ iki \"u\c cgendir\\
\gk{ABG} ve \gk{BGD},\\
iki \gk{ABG} ve \gk{BGA} a\c c\i lar\i\\
iki \gk{BGD} ve \gk{GBD} a\c c\i lar\i na\\
e\c sit olan,\\
her biri birine,\\
ve bir kenar\i, bir kenar\i na e\c sit olan,\\
e\c sit a\c c\i lar\i n yan\i nda olan,\\
onlar\i n ortak \gk{BG};\\
\ara\ kalan kenarlar\i\ da\\
kalan kenarlar\i na e\c sit olacaklar,\\
her biri birine,\\
ve kalan a\c c\i\\
kalan a\c c\i ya;\\
\ara\ e\c sittir\\
 \gk{AB} kenar\i\ \gk{GD}'ya,\\
ve \gk{AG}, \gk{BD}'ya,\\
ve e\c sittir\\
\gk{BAG} a\c c\i s\i, \gk{GDB}'ya.\\
Ve e\c sit oldu\u gundan\\
\gk{ABG} a\c c\i s\i, \gk{BGD}'ya,\\
ve \gk{GBD}, \gk{AGB} a\c c\i s\i na,\\
\ara\ b\"ut\"un \gk{ABD},\\
b\"ut\"un \gk{AGD}'ya e\c sittir.\\
Ve g\"osterilmi\c sti\\
\gk{BAG} da, \gk{GDB}'ya e\c sit.
}

\newparsen{
T~wn >'ara\eix{\gr{>'ara}} parallhlogr'ammwn qwr'iwn\\
a<i >apenant'ion pleura'i te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} gwn'iai\\
{}>'isai >all'hlaic e>is'in.
}
{
\Ara, paralelkenar alanlar\i n\\
hem kar\c s\i t kenarlar\i\ hem de a\c c\i lar\i,\\
birbirine e\c sittir.
}

\newparsen{
L'egw d'h, <'oti\\
\kai%
\eix{\gr{ka`i}} <h di'ametroc a>ut`a d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}.
}
{
\Dee\ \legohoti\\
k\"o\c segen de onlar\i\ ikiye b\"oler.
}

\newparsen{
>epe`i%
\eix{\gr{>epe'i}} g`ar >'ish >est`in <h AB t~h| GD,\\
koin`h d`e <h BG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i AB, BG\\
dus`i ta~ic GD, BG >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ABG\\
gwn'ia| t~h| <up`o BGD {}>'ish.\\
\kai%
\eix{\gr{ka`i}} b'asic >'ara\eix{\gr{>'ara}} <h AG\\
t~h| DB >'ish.\\
\kai%
\eix{\gr{ka`i}} t`o ABG [>'ara\eix{\gr{>'ara}}] tr'igwnon\\
t~w| BGD trig'wnw|
{}>'ison >est'in.
}
{
\Gar\ \gk{AB}, \gk{GD}'ya e\c sit oldu\u gundan,\\
ve \gk{BG} ortak oldu\u gundan,\\
\dee\ \gk{AB} ve \gk{BG} ikilisi\\
\gk{GD} ve \gk{BG} ikilisine e\c sittir,\\
her biri birine;\\
ve \gk{ABG} a\c c\i s\i,\\
\gk{BGD} a\c c\i s\i na e\c sittir.\\
\Ara\ \gk{AG} taban\i\ da,\\
\gk{DB}'ya e\c sittir.\\
\Ara\ \gk{ABG} \"u\c cgeni de\\
\gk{BGD} \"u\c cgenine
e\c sittir.
}

\parsen{
Therefore the \gk{BG} diameter cuts in two\\
the \gk{ABGD} parallelogram;\\
\myqed
}
{
<H >'ara\eix{\gr{>'ara}} BG di'ametroc d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}\\
t`o ABGD parallhl'ogrammon;\\
<'oper >'edei de~ixai.
}
{
\Ara\ \gk{BG} k\"o\c segeni ikiye b\"oler\\
\gk{ABGD} paralelkenar\i n\i;\\
\ozqed.
}
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\rput(0.10828125,-1.236875){\gk G}
\end{pspicture} 
}
\end{center}


\end{proposition}

\begin{proposition}{35}%Proposition I.35

\newparsen{
T`a parallhl'ogramma\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic
a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in.
}
{
Paralelkenarlar\\
ayn\i\ tabanda olan\\
ve ayn\i\ paralellerde,\\
birbirine e\c sittir.
}

\newparsen{
>'Estw\\
parallhl'ogramma t`a ABGD, EBGZ\\
{}>ep`i t~hc a>ut~hc b'asewc t~hc BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic ta~ic AZ, BG;
}
{
Olsun\\
paralelkenarlar \gk{ABGD} ve \gk{EBGD},\\
ayn\i\ \gk{GB} taban\i nda,\\
ve ayn\i\ \gk{AZ} ve \gk{BG} paralellerinde.
}

\newparsen{
l'egw, <'oti\\
{}>'ison >est`i t`o ABGD\\
t~w| EBGZ parallhlogr'ammw|.
}
{
\Legohoti\\
\gk{ABGD} e\c sittir\\
\gk{EBGZ} paralelkenar\i na.
}

\newparsen{
>Epe`i%
\eix{\gr{>epe'i}} g`ar
parallhl'ogramm'on >esti\\
t`o ABGD,\\
{}>'ish >est`in <h AD t~h| BG.\\
di`a\eix{\gr{di'a}} t`a a>ut`a d`h%
\eix{\gr{d'h}}\\
\kai%
\eix{\gr{ka`i}} <h EZ t~h| BG >estin >'ish;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} <h AD t~h| EZ >estin >'ish;\\
\kai%
\eix{\gr{ka`i}} koin`h <h DE;\\
<'olh >'ara\eix{\gr{>'ara}} <h AE\\
<'olh| t~h| DZ >estin >'ish.\\
{}>'esti d`e \kai%
\eix{\gr{ka`i}} <h AB t~h| DG >'ish;\\
d'uo d`h%
\eix{\gr{d'h}} a<i EA, AB\\
d'uo ta~ic ZD, DG >'isai e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o ZDG\\
gwn'ia| t~h| <up`o EAB >estin >'ish\\
<h >ekt`oc t~h| >ent'oc;\\
b'asic >'ara\eix{\gr{>'ara}} <h EB\\
b'asei t~h| ZG >'ish >est'in,\\
\kai%
\eix{\gr{ka`i}} t`o EAB tr'igwnon\\
t~w| DZG trig'wnw| >'ison >'estai;\\
koin`on >afh|r'hsjw%
\eix{\gr{<airew}!\gr{>afairew}!\gr{>afh"|r'hsjw} 3rd sg perf imperat mp} t`o DHE;\\
loip`on >'ara\eix{\gr{>'ara}} t`o ABHD trap'ezion\\
loip~w| t~w| EHGZ trapez'iw| >est`in >'ison;\\
koin`on proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp}\\
t`o HBG tr'igwnon;\\
<'olon >'ara\eix{\gr{>'ara}} t`o ABGD parallhl'ogrammon\\
<'olw| t~w| EBGZ parallhlogr'ammw| >'ison >est'in.
}
{
\Gar\ paralelkenar oldu\u gundan\\
\gk{ABGD},\\
\gk{AD}, \gk{BG}'ya e\c sittir.\\
\Diatauta\ \dee\\
\gk{EZ} da, \gk{BG}'ya e\c sittir;\\
\hoste\ \gk{AD} da \gk{EZ}'ya e\c sittir;\\
ve \gk{DE} ortakt\i r;\\
\ara\ b\"ut\"un \gk{AE},\\
b\"ut\"un \gk{DZ}'ya e\c sittir.\\
\gk{AB} da  \gk{DG}'ya e\c sittir.\\
\Dee\ \gk{EA} ve \gk{AB} ikilisi\\
 \gk{ZD} ve \gk{DG} ikilisine e\c sittir\\
her biri birine;\\
ve \gk{ZDG} a\c c\i s\i\ da\\
 \gk{EAB} a\c c\i s\i na e\c sittir,\\
d\i\c s\ a\c c\i, i\c c\ a\c c\i ya;\\
\ara\ \gk{EB} taban\i\\
\gk{ZG} taban\i na e\c sittir,\\
ve \gk{EAB} \"u\c cgeni\\
 \gk{DZG} \"u\c cgenine
e\c sit olacak;\\
ortak \gk{DHE} ayr\i lm\i\c s olsun;\\
\ara\ kalan \gk{ABHD} yamu\u gu\footnotemark\\
kalan \gk{EHGZ} yamu\u guna e\c sittir;\\
ortak olarak eklenmi\c s\ olsun\\
\gk{HBG} \"u\c cgeni;\\
\ara\ b\"ut\"un \gk{ABGD} paralelkenar\i,\lli\\
b\"ut\"un \gk{EBGZ} paralelkenar\i na e\c sittir.
}\footnotetext{Yani \emph{trapezion.}}

\parsen{
Therefore parallelograms\\
on the same base being\\
and in the same parallels\\
equal to one another are;\\
\myqed
}
{
T`a >'ara\eix{\gr{>'ara}} parallhl'ogramma\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ 
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara\ paralelkenarlar;\\
ayn\i\ tabanda olan\\
ve ayn\i\ paralellerde olanlar,\\
birbirine e\c sittir;\\
\ozqed.
}
\begin{center}
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}
\end{center}
\end{proposition}

\begin{proposition}{36}%Proposition I.36

\parsen{
Parallelograms\\
that are on equal bases\\
and in the same parallels\\
are equal to one another.
}
{
T`a parallhl'ogramma\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in.
}
{
Paralelkenarlar\\
e\c sit tabanlarda olan\\
ve ayn\i\ paralellerde,\\
birbirine e\c sittir.
}


\parsen{
Let there be\\
parallelograms\\
\gk{ABGD} and \gk{EZHJ}\\
on equal bases,\\
\gk{BG} and \gk{ZH},\\
and in the same parallels,\\
\gk{AJ} and \gk{BH}.
}
{
>'Estw\\
parallhl'ogramma t`a ABGD, EZHJ\\
{}>ep`i >'iswn b'asewn >'onta t~wn BG, ZH\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic ta~ic AJ, BH;
}
{
Olsun\\
paralelkenarlar \gk{ABGD} ve \gk{EZHJ}\\
e\c sit \gk{BG} ve \gk{ZH} tabanlar\i nda,\\
ve ayn\i\ \gk{AJ} ve \gk{BH} paralellerinde.
}


\parsen{
I say that\\
equal is\\
parallelogram \gk{ABGD}\\
to \gk{EZHJ}.
}
{
l'egw, <'oti\\
{}>'ison >est`i t`o ABGD parallhl'ogrammon\\
t~w| EZHJ.
}
{
\Legohoti\\
\gk{ABGD} paralelkenar\i\ e\c sittir\\
\gk{EZHJ}'ya.
}


\parsen{
For, suppose have been joined\\
\gk{BE} and \gk{GJ}.
}
{
>Epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp} g`ar\\
a<i BE, GJ.
}
{
\Gar\ birle\c stirilmi\c s\ olsun\\
\gk{BE} ile \gk{GJ}.
}


\parsen{
And since equal are \gk{BG} and \gk{ZH},\\
but \gk{ZH} to \gk{EJ} is equal,\\
therefore also \gk{BG} to \gk{EJ} is equal.\\
And [they] are also parallel.\\
Also \gk{EB} and \gk{JG} join them.\\
And [\strgt s] that join equals and parallels in the same parts\\
are equal and parallel.\\
{}[Also therefore \gk{EB} and \gk{HJ}\\are equal and parallel.]\\
Therefore a parallelogram is \gk{EBGJ}.\\
And it is equal to \gk{ABGD}.\\
For it has the same base as it,\\
\gk{BG},\\
and in the same parallels\\
as it it is,
\gk{BG} and \gk{AJ}.\\
For the same [reason] then,\\
also \gk{EZHJ} to it, [namely] \gk{EBGJ},\\
is equal;\\
so that parallelogram \gk{ABGD}\\
to \gk{EZHJ} is equal.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in <h BG t~h| ZH,\\
{}>all`a%
\eix{\gr{>all'a}} <h ZH t~h| EJ >estin >'ish,\\
\kai%
\eix{\gr{ka`i}} <h BG >'ara\eix{\gr{>'ara}} t~h| EJ >estin >'ish.\\ 
e>is`i d`e \kai%
\eix{\gr{ka`i}} par'allhloi.\\
\kai%
\eix{\gr{ka`i}} >epizeugn'uousin%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}} a>ut`ac a<i EB, JG;\\
a<i d`e t`ac >'isac te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} parall'hlouc\\
{}>ep`i t`a a>ut`a m'erh >epizeugn'uousai%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epizeugn'uwn} part pres act}\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhlo'i e>isi\\
{}[\kai%
\eix{\gr{ka`i}} a<i EB, JG >'ara\eix{\gr{>'ara}}\\
{}>'isai t'e e>isi \kai%
\eix{\gr{ka`i}} par'allhloi].\\
parallhl'ogrammon >'ara\eix{\gr{>'ara}} >est`i t`o EBGJ.\\
ka'i >estin >'ison t~w| ABGD;\\
b'asin te%
\eix{\gr{te}} g`ar a>ut~w| t`hn a>ut`hn >'eqei%
\eix{\gr{>'eqw}}
t`hn BG,\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic
{}>est`in a>ut~w| ta~ic BG, AJ.\\
di`a\eix{\gr{di'a}} t`a a>ut`a d`h%
\eix{\gr{d'h}}\\
\kai%
\eix{\gr{ka`i}} t`o EZHJ\\
t~w| a>ut~w| t~w| EBGJ >estin >'ison;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} t`o ABGD parallhl'ogrammon\\
t~w| EZHJ >estin >'ison.
}
{
Ve e\c sit oldu\u gundan \gk{BG} ile \gk{ZH},\\
ama \gk{ZH}, \gk{EJ}'ya e\c sit oldu\u gundan,\\
\ara\ \gk{BG} da, \gk{EJ}'ya e\c sittir.\\
Ve paraleldirler de.\\
Ve \gk{EB} ve \gk{JG} onlar\i\ birle\c stirir.\\
Ve hem e\c sit hem paraleller\\
ayn\i\ tarafta birle\c stirenler\\
hem e\c sit hem paraleldir.\\
{}[Ve \ara\ \gk{EB} ve \gk{HJ},\\
hem e\c sit hem paraleldir.]\\
\Ara\ \gk{EBGJ} bir paralelkenard\i r.\lli\\
Ve e\c sittir \gk{ABGD}'ya.\\
\Gar\ onunla ayn\i\ \gk{BG} taban\i\ vard\i r,\lli\\
ve onunla ayn\i\ \gk{BG} ve \gk{AJ} paralellerindedir.\\
\Diatauta\ \dee,\\
\gk{EZHJ}  da,\\
ayn\i\ \gk{EBGJ}'ya e\c sittir;\\
\hoste\ \gk{ABGD} paralelkenar\i\ da,\\
\gk{EZHJ}'ya e\c sittir.
}

\parsen{
Therefore parallelograms\\
that are on equal bases\\
and in the same parallels\\
are equal to one another;\\
\myqed
}
{
T`a  >'ara\eix{\gr{>'ara}} parallhl'ogramma\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara\ paralelkenarlar\\
e\c sit tabanlarda olan\\
ve ayn\i\ paralellerde,\\
birbirine e\c sittir;\\
\ozqed.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
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}
\end{center}
\end{proposition}

\begin{proposition}{37}%Proposition I.37

\newparsen{
T`a tr'igwna\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in.
}
{
\"U\c cgenler\\
ayn\i\ tabanda olan\\
ve ayn\i\ paralellerde,\\
birbirine e\c sittir.
}

\newparsen{
>'Estw\\
tr'igwna t`a ABG, DBG\\
{}>ep`i t~hc a>ut~hc b'asewc t~hc BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
ta~ic AD, BG;
}
{
Olsun\\
\"u\c cgenler \gk{ABG} ve \gk{DBG},\\
ayn\i\ \gk{BG} taban\i nda\\
ve ayn\i\ paralellerinde\\
{}[yani] \gk{AD} ve \gk{BG}.
}

\newparsen{
l'egw, <'oti\\
{}>'ison >est`i
t`o ABG tr'igwnon\\
t~w| DBG trig'wnw|.
}
{
\Legohoti\\
\gk{ABG} \"u\c cgeni, e\c sittir\\
\gk{DBG} \"u\c cgenine.
}

\newparsen{
>Ekbebl'hsjw%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjw} 3rd sg perf imperat mp}\\
<h AD >ef> <ek'atera t`a m'erh\\
{}>ep`i t`a E, Z,\\
\kai%
\eix{\gr{ka`i}} di`a m`en%
\eix{\gr{m'en}} to~u B\\
t~h| GA par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h BE,\\
di`a d`e to~u G\\
t~h| BD par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h GZ.
}
{
Uzat\i lm\i\c s\ olsun\\
\gk{AD} do\u grusu, her iki kenarda,\\
\gk E ve \gk Z noktalar\i na,\\
ve \gk B'dan,\\
\gk{GA}'ya paralel\\
\gk{BE} ilerletilmi\c s olsun,\\
ve  \gk G'dan\\
 \gk{BD}'ya paralel\\
 \gk{GZ} ilerletilmi\c s olsun.
}

\newparsen{
parallhl'ogrammon >'ara\eix{\gr{>'ara}}\\
{}>est`in <ek'ateron t~wn EBGA, DBGZ;\\
ka'i e>isin >'isa;\\
{}>ep'i te%
\eix{\gr{te}} g`ar t~hc a>ut~hc b'ase'wc e>isi t~hc BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic ta~ic BG, EZ;\\
ka'i >esti to~u m`en%
\eix{\gr{m'en}} EBGA parallhlogr'ammou <'hmisu\\
t`o ABG tr'igwnon;\\
<h g`ar AB di'ametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act};\\
to~u d`e DBGZ parallhlogr'ammou\\
<'hmisu t`o DBG tr'igwnon;\\
<h g`ar DG di'ametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}.\\
{}[t`a d`e t~wn >'iswn <hm'ish\\
{}>'isa >all'hloic >est'in].\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i\\
t`o ABG tr'igwnon t~w| DBG trig'wnw|.
}
{
\Ara\ paralelkenard\i r\\
birer \gk{EBGA} ile \gk{DBGZ};\\
ve [bunlar] e\c sittir;\\
\gar\ hem ayn\i\ \gk{BG} taban\i nda\lli\\
hem ayn\i\ \gk{BG} ve \gk{EZ} paralellerinde;\lli\\
ve \gk{EBGA} paralelkenar\i n\i n yar\i s\i,\lli\\
\gk{ABG} \"u\c cgenidir,\\
\gar\ 
\gk{AB} k\"o\c segeni onu ikiye b\"oler;\\
ve \gk{DBGZ} paralelkenar\i n\i n\\
yar\i s\i, \gk{DBG} \"u\c cgenidir,\\
\gar\ \gk{DG} k\"o\c segeni onu ikiye b\"oler.\\
{}[Ve e\c sitlerin yar\i lar\i\\
birbirine e\c sittir.]\\
\Ara\ e\c sittir\\
\gk{ABG} \"u\c cgeni \gk{DBG} \"u\c cgenine.
}


\newparsen{
T`a >'ara\eix{\gr{>'ara}} tr'igwna\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara\ \"u\c cgenler\\
ayn\i\ tabanda olan\\
ve ayn\i\ paralellerde,\\
birbirine e\c sittir;\\
\ozqed.
}
\begin{center}
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{
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\rput(4.5735936,-1.6515625){\gk G}
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}
\end{center}

\end{proposition}

\begin{proposition}{38}%Proposition I.38

\parsen{
Triangles\\
that are on equal bases\\
and in the same parallels\\
are equal to one another.
}
{
T`a tr'igwna\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in.
}
{
\"U\c cgenler\\
e\c sit tabanlarda olan\\
ve ayn\i\ paralelerde,\\
birbirine e\c sittir.
}

\parsen{
Let there be\\
triangles \gk{ABG} and \gk{DEZ}\\
on equal bases \gk{BG} and \gk{EZ}\\
and in the same parallels\\
\gk{BZ} and \gk{AD}.
}
{
>'Estw\\
tr'igwna t`a ABG, DEZ\\
{}>ep`i >'iswn b'asewn t~wn BG, EZ\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic ta~ic BZ, AD;\lli
}
{
Olsun\\
\"u\c cgenler \gk{ABG} ve \gk{DEZ}\\
e\c sit \gk{BG} ve \gk{EZ} tabanlar\i nda\\
ve ayn\i\ \gk{BZ} ve \gk{AD} paralellerinde.
}


\parsen{
I say that\\
equal is\\
triangle \gk{ABG}\\
to triangle \gk{DEZ}.
}
{
l'egw, <'oti\\
{}>'ison >est`i t`o ABG tr'igwnon\\
t~w| DEZ trig'wnw|.
}
{
\Legohoti\\
\gk{ABG} \"u\c cgeni,
e\c sittir\\
\gk{DEZ} \"u\c cgenine.
}

\parsen{
For, suppose has been extended\\
\gk{AD} on both sides to \gk H and \gk J,\\
and through \gk B,\\
parallel to \gk{GA},\\
has been drawn \gk{BH},\\
and through \gk Z,\\
parallel to \gk{DE},\\
has been drawn \gk{ZJ}.
}
{
>Ekbebl'hsjw%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjw} 3rd sg perf imperat mp} g`ar
<h AD\\
{}>ef> <ek'atera t`a m'erh >ep`i t`a H, J,\\
\kai%
\eix{\gr{ka`i}} di`a m`en%
\eix{\gr{m'en}} to~u B\\
t~h| GA par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h BH,\\
di`a d`e to~u Z\\
t~h| DE par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h ZJ.
}
{
\Gar\ \gk{AD} uzat\i lm\i\c s\ olsun\\
her iki tarafta \gk H ve \gk J'ya,\\
ve \gk B'dan,\\
\gk{GA}'ya paralel,\\
\gk{BH} ilerletilmi\c s olsun,\\
ve \gk Z'dan,\\
\gk{DE}'a paralel,\\
\gk{ZJ} ilerletilmi\c s olsun.\\
}

\parsen{
Therefore a parallelogram\\
is either of \gk{HBGA} and \gk{DEZJ};\\
and \gk{HBGA} [is] equal to \gk{DEZJ};\\
for they are on equal bases,\\
\gk{BG} and \gk{EZ},\\
and in the same parallels,\\
\gk{BZ} and \gk{HJ};\\
and [it] is\\
of the parallelogram \gk{HBGA}\\
half\\
---the triangle \gk{ABG}.\\
For the diameter \gk{AB} cuts it in two;\\
and of the parallelogram \gk{DEZJ}\\
half\\
---the triangle \gk{ZED};\\
for the diameter \gk{DZ} cuts it in two.\\
{}[And halves of equals\\
are equal to one another.]\\
Therefore equal is\\
the triangle \gk{ABG} to the triangle \gk{DEZ}.
}
{
 parallhl'ogrammon >'ara\eix{\gr{>'ara}}\\
{}>est`in <ek'ateron t~wn HBGA, DEZJ;\\
\kai%
\eix{\gr{ka`i}} >'ison t`o HBGA t~w| DEZJ;\\
{}>ep'i te%
\eix{\gr{te}} g`ar >'iswn b'ase'wn e>isi t~wn BG, EZ\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic ta~ic BZ, HJ;\\
ka'i >esti to~u m`en%
\eix{\gr{m'en}} HBGA parallhlogr'ammou <'hmisu\\
t`o ABG tr'igwnon.\\
<h g`ar AB di'ametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act};\\
to~u d`e DEZJ parallhlogr'ammou <'hmisu\\
t`o ZED tr'igwnon;\\
<h g`ar DZ d'iametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act}\\
{}[t`a d`e t~wn >'iswn <hm'ish\\
{}>'isa >all'hloic >est'in].\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i
t`o ABG tr'igwnon\\
t~w| DEZ trig'wnw|.
}
{
\Ara\ paralelkenard\i r\\
birer \gk{HBGA} ile \gk{DEZJ};\\
ve \gk{HBGA}, \gk{DEZJ}'ya e\c sittir;\\
\gar\ hem e\c sit \gk{BG} ve \gk{EZ} tabanlar\i nda,\lli\\
hem ayn\i\ \gk{BZ} ve \gk{HJ} paralellerinde;\lli\\
ve \gk{HBGA} paralelkenar\i n\i n yar\i s\i,\lli\\
\gk{ABG} \"u\c cgenidir.\\
\Gar\ \gk{AB} k\"o\c segeni onu ikiye b\"oler;\\
ve \gk{DEZJ} paralelkenar\i n\i n yar\i s\i,\lli\\
\gk{ZED} \"u\c cgenidir;\\
\gar\ \gk{DZ} k\"o\c segeni onu ikiye b\"oler.\\
{}[Ve e\c sitlerin yar\i lar\i,\\
birbirine e\c sittir.]\\
\Ara\ \gk{ABG} \"u\c cgeni e\c sittir\\
\gk{DEZ} \"u\c cgenine.
}

\parsen{
Therefore triangles\\
that are on equal bases\\
and in the same parallels\\
are equal to one another;\\
\myqed
}
{
T`a >'ara\eix{\gr{>'ara}} tr'igwna\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic\\
{}>'isa >all'hloic >est'in;\\
<'oper >'edei de~ixai.  
}
{
\Ara\ \"u\c cgenler\\
e\c sit tabanlarda olan\\
ve ayn\i\ paralelerde,\\
birbirine e\c sittir;\\
\ozqed.
}

\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
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\end{center}
\end{proposition}

\begin{proposition}{39}%Proposition I.39

\parsen{
Equal triangles\\
that are on the same base\\
and in the same parts\\
are also in the same parallels.
}
{
T`a >'isa tr'igwna\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in.
}
 {
E\c sit \"u\c cgenler\\
ayn\i\ tabanda olan\\
ve ayn\i\ taraf\i nda,\\
ayn\i\ paralellerdedir de.
}


\parsen{
Let there be\\
equal triangles \gk{ABG} and \gk{DBG},\\
being on the same base\\
and on the same side of \gk{BG}.
}
{
>'Estw\\
{}>'isa tr'igwna t`a ABG, DBG\\
{}>ep`i t~hc a>ut~hc b'asewc >'onta
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh t~hc BG;
}
{
Olsun\\
e\c sit \"u\c cgenleri \gk{ABG} ve \gk{DBG}\\
ayn\i\ \gk{BG} taban\i nda
ve ayn\i\ taraf\i nda olan.
}

\parsen{
I say that\\
they are also in the same parallels.
}
{
[l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in.]
}
{
{}[\Legohoti\\
ayn\i\ paralellerdedirler de.]\footnotemark
}\footnotetext{Heath'in notuna \cite[I.337]{MR17:814b} bak\i n\i z.}

\parsen{
For suppose has been joined \gk{AD}.
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} [g`ar] <h AD; 
}
{
{}[\Gar]\footnotemark \gk{AD} birle\c stirilmi\c s\ olsun.
}

\parsen{
I say that\\
parallel is \gk{AD} to \gk{BG}.
}
{
l'egw, <'oti\\
par'allhl'oc >estin <h AD t~h| BG.
}
{
\Legohoti\\
paraleldir \gk{AD}, \gk{BG} taban\i na.
}

\parsen{
For if not,\\
suppose there has been drawn\\
through the point \gk A\\
parallel to the \strgt\ \gk{BG}\\
\gk{AE},\\
and there has been joined \gk{EG}.\\
Equal therefore is\\
the triangle \gk{ABG}\\
to the triangle \gk{EBG};\\
for on the same base\\
as it it is, \gk{BG},\\
and in the same parallels.\\
But \gk{ABG} is equal to \gk{DBG}.\\
Also therefore \gk{DBG} to \gk{EBG} is equal,\\
the greater to the less;\\
which is impossible.\\
Therefore is not parallel \gk{AE} to \gk{BG}.\\
Similarly then we shall show that\\
neither is any other but \gk{AD};\\
therefore \gk{AD} is parallel to \gk{BG}.
}
{
E>i g`ar m'h,\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp}\\
di`a to~u A shme'iou\\
t~h| BG e>uje'ia| par'allhloc\\
<h AE,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h EG.\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i\\
t`o ABG tr'igwnon\\
t~w| EBG trig'wnw|;\\
{}>ep'i te%
\eix{\gr{te}} g`ar t~hc a>ut~hc b'ase'wc >estin a>ut~w| t~hc BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic.\\
{}>all`a%
\eix{\gr{>all'a}} t`o ABG t~w| DBG >estin >'ison;\\
\kai%
\eix{\gr{ka`i}} t`o DBG >'ara\eix{\gr{>'ara}} t~w| EBG >'ison >est`i\\
t`o me~izon t~w| >el'assoni;\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}};\\
o>uk >'ara\eix{\gr{>'ara}} par'allhl'oc >estin\\
<h AE t~h| BG.\\
<omo'iwc d`h%
\eix{\gr{d'h}} de'ixomen%
\eix{\gr{de'iknumi}!\gr{de'ixomen} 1st pl fut ind act},\\
<'oti\\
o>ud> >'allh tic pl`hn t~hc AD;\\
<h AD >'ara\eix{\gr{>'ara}} t~h| BG >esti par'allhloc.
}
{
\Gar\ e\u ger de\u gil ise,\\
ilerletilmi\c s\ olsun\\
\gk A noktas\i ndan\\
\gk{BG} do\u grusuna paralel\\
\gk{AE},\\
ve \gk{EG} birle\c stirilmi\c s olsun.\\
E\c sittir \ara\\
\gk{ABG} \"u\c cgeni,\\
\gk{EBG} \"u\c cgenine;\\
\gar\ hem onunla ayn\i\ \gk{BG} taban\i nda,\lli\\
hem ayn\i\ paralellerdedir.\\
Ama \gk{ABG}, \gk{DBG}'ya e\c sittir.\\
Ve \ara\ \gk{DBG}, \gk{EBG}'ya e\c sittir,\\
b\"uy\"uk k\"u\c c\"u\u ge;\\
\imkansiz.\\
\Ara\ paralel de\u gildir\\
\gk{AE}, \gk{BG}'ya.\\
Benzer \c sekilde \dee\ g\"osterece\u giz\\
ki\\
\gk{AD} d\i\c s\i ndakiler de [paralel] de\u gildir;\\
\ara\ \gk{AD}, \gk{BG}'ya paraleldir.
}

\parsen{
Therefore equal triangles\\
that are on the same base\\
and in the same parts\\
are also in the same parallels;\\
\myqed
}
{
T`a  >'ara\eix{\gr{>'ara}} >'isa tr'igwna\\
t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara\ e\c sit \"u\c cgenler\\
ayn\i\ tabanda olan\\
ve onun ayn\i\ taraf\i nda,\\
ayn\i\ paralellerdedirler de;\\
\ozqed.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
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\rput(5.0940623,1.543125){\gk D}
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}
\end{center}
\end{proposition}

\begin{proposition}{40}%Proposition I.40

(Bu \"onerme, \"Oklid'in orijinal metne bir il\^avedir.  Heath'in \cite[I.338]{MR17:814b} notuna bak\i n\i z.)

\parsen{
Equal triangles\\
that are on equal bases\\
and in the same parts\\
are also in the same parallels.
}
{
T`a >'isa tr'igwna\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in.
}
{
E\c sit \"u\c cgenler,\\
e\c sit tabanlarda\\
ve ayn\i\ tarafta olan,\\
ayn\i\ paralelerdedirler de.
}
\parsen{
Let there be\\
equal triangles \gk{ABG} and \gk{GDE},\\
on equal bases \gk{BG} and \gk{GE},\\
and in the same parts.
}
{
>'Estw\\
{}>'isa tr'igwna t`a ABG, GDE\\
{}>ep`i >'iswn b'asewn t~wn BG, GE\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh.
}
{
Olsun\\
e\c sit \"u\c cgenler \gk{ABG} ve \gk{GDE},\\
e\c sit \gk{BG} ve \gk{GE} tabanlar\i nda,\\
ve ayn\i\ tarafta olan.
}

\parsen{
I say that\\
they are also in the same parallels.
}
{
l'egw, <'oti\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in.
}
{
\Legohoti\\
ayn\i\ paralellerdedirler de.
}

\parsen{
For suppose \gk{AD} has been joined.
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} g`ar <h AD; 
}
{
\Gar\ \gk{AD} birle\c stirilmi\c s olsun.
}

\parsen{
I say that\\
parallel is \gk{AD} to \gk{BE}.
}
{
l'egw, <'oti\\
par'allhl'oc >estin <h AD t~h| BE.
}
{
\Legohoti\\
paraleldir \gk{AD}, \gk{BE} do\u grusuna.
}

\parsen{
For if not,\\
suppose there has been drawn\\
through the point \gk A,\\
parallel to \gk{BE},\\
\gk{AZ},\\
and there has been joined \gk{ZE}.\\
Equal therefore is\\
the triangle \gk{ABG}\\
to the triangle \gk{ZGE};\\
for they are on equal bases,\\
\gk{BG} and \gk{GE},\\
and in the same parallels,\\
\gk{BE} and \gk{AZ}.\\
But the triangle \gk{ABG}\\
is equal to the [triangle] \gk{DGE};\\
also therefore the [triangle] \gk{DGE}\\
is equal to the triangle \gk{ZGE},\\
the greater to the less;\\
which is impossible.\\
Therefore is not parallel \gk{AZ} to \gk{BE}.\\
Similarly then we shall show that\\
neither is any other but \gk{AD};\\
therefore \gk{AD} to \gk{BE} is parallel.
}
{
E>i g`ar m'h,\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp}\\
di`a to~u A\\
t~h| BE par'allhloc\\
<h AZ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h ZE.\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i\\
t`o ABG tr'igwnon\\
t~w| ZGE trig'wnw|;\\
{}>ep'i te%
\eix{\gr{te}} g`ar >'iswn b'ase'wn e>isi
t~wn BG, GE\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic
ta~ic BE, AZ.\\
{}>all`a%
\eix{\gr{>all'a}} t`o ABG tr'igwnon\\
{}>'ison >est`i t~w| DGE  [tr'igwnw|];\\
\kai%
\eix{\gr{ka`i}} t`o DGE >'ara\eix{\gr{>'ara}} [tr'igwnon]\\
{}>'ison >est`i t~w| ZGE trig'wnw|\\
t`o me~izon t~w| >el'assoni;\\
<'oper >est`in >ad'unaton\eix{\gr{<'oper >est`in >ad'unaton}};\\
o>uk >'ara\eix{\gr{>'ara}} par'allhloc\\
<h AZ t~h| BE.\\
<omo'iwc d`h%
\eix{\gr{d'h}} de'ixomen%
\eix{\gr{de'iknumi}!\gr{de'ixomen} 1st pl fut ind act},\\
<'oti\\
o>ud> >'allh tic pl`hn t~hc AD;\\
<h AD >'ara\eix{\gr{>'ara}} t~h| BE >esti par'allhloc.
}
{
\Gar\ e\u ger de\u gil ise,\\
ilerletilmi\c s olsun\\
\gk A noktas\i ndan,\\
\gk{BE}'a paralel,\\
\gk{AZ},\\
ve \gk{ZE} birle\c stirilmi\c s olsun.\\
\Ara\ e\c sittir\\
\gk{ABG} \"u\c cgeni\\
\gk{ZGE} \"u\c cgenine;\\
\gar\ hem e\c sit \gk{BG} ve \gk{GE} tabanlar\i nda,\lli\\
hem ayn\i\ \gk{BE} ve \gk{AZ} paralellerindedir.\\
Ama \gk{ABG} \"u\c cgeni,\\
\gk{DGE} \"u\c cgenine e\c sittir;\\
ve \ara\ \gk{DGE} \"u\c cgenini\\
\gk{ZGE} \"u\c cgenine e\c sittir,\\
b\"uy\"uk k\"u\c c\"u\u ge;\\
\imkansiz.\\
\Ara\ paralel de\u gildir\\
\gk{AZ}, \gk{BE}'a.\\
Benzer \c sekilde \dee\ g\"osterece\u giz\\
ki\\
\gk{AD} d\i\c s\i ndakiler de [paralel] de\u gildir;\\
\ara\ \gk{AD}, \gk{BE}'a paraleldir.
}

\parsen{
Therefore equal triangles\\
that are on equal bases\\
and in the same parts\\
are also in the same parallels;\\
\myqed
}
{
T`a >'ara\eix{\gr{>'ara}} >'isa tr'igwna\\
t`a >ep`i >'iswn b'asewn >'onta\\
\kai%
\eix{\gr{ka`i}} >ep`i t`a a>ut`a m'erh\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara\ e\c sit \"u\c cgenler\\
e\c sit tabanlarda olan\\
ve ayn\i\ tarafta,\\
ayn\i\ paralelerdedir de;\\
\ozqed.
}
\begin{center}
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}
\end{center}
\end{proposition}

\begin{proposition}{41}%Proposition I.41

\parsen{
If a parallelogram\\
have the same base as a triangle,\\
and be in the same parallels,\\
double is\\
the parallelogram of the triangle.
}
{
>E`an parallhl'ogrammon\\
 trig'wnw|\\
b'asin te%
\eix{\gr{te}} >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act} t`hn a>ut`hn\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >~h|,\\
dipl'asi'on >est'i\\
t`o parallhl'ogrammon to~u trig'wnou.
}
{
E\u ger bir paralelkenar\\
bir \"u\c cgenle\\
hem ayn\i\ tabana sahipse,\\
hem ayn\i\ paralelerdeyse,\\
iki kat\i d\i r\\
paralelkenar, \"u\c cgenin.
}

\parsen{
For, the parallelogram \gk{ABGD}\\
as the triangle \gk{EBG},\\
---suppose it has the same base, \gk{BG},\\
and is in the same parallels,\\
\gk{BG} and \gk{AE}.
}
{
Parallhl'ogrammon g`ar t`o ABGD\\
trig'wnw| t~w| EBG\\
b'asin te%
\eix{\gr{te}} >eq'etw t`hn a>ut`hn t`hn BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >'estw
ta~ic BG, AE;
}
{
\Gar\ \gk{ABGD} paralelkenar\i,\\
\gk{EBG} \"u\c cgeniyle\\
hem ayn\i\ \gk{BG} taban\i na sahip olsun,\\
hem ayn\i\ \gk{BG} ve \gk{AE} paralellerinde olsun.
}

\parsen{
I say that\\
double is\\
the parallelogram \gk{ABGD}\\
of the triangle \gk{BEG}.
}
{
l'egw, <'oti\\
dipl'asi'on >esti\\
t`o ABGD parallhl'ogrammon\\
to~u BEG trig'wnou.
}
{
\Legohoti\\
iki kat\i d\i r\\
\gk{ABGD} paralelkenar\i,\\
\gk{BEG} \"u\c cgeninin.
}

\parsen{
For, suppose \gk{AG} has been joined.
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} g`ar <h AG. 
}
{
\Gar\ \gk{AG} birle\c stirilmi\c s olsun.
}

\parsen{
Equal is the triangle \gk{ABG}\\
to the triangle \gk{EBG};\\
for it is on the same base as it,\\
\gk{BG},\\
and in the same parallels,\\
\gk{BG} and \gk{AE}.\\
But the parallelogram \gk{ABGD}\\
is double of the triangle \gk{ABG};\\
for the diameter \gk{AG} cuts it in two;\\
so that the parallelogram \gk{ABGD}\\
also of the triangle \gk{EBG} is double.
}
{
>'ison d'h >esti t`o ABG tr'igwnon\\
t~w| >EBG trig'wnw|;\\
{}>ep'i te%
\eix{\gr{te}} g`ar t~hc a>ut~hc b'ase'wc >estin a>ut~w|
t~hc BG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic
ta~ic BG, AE.\\
{}>all`a%
\eix{\gr{>all'a}} t`o >ABGD parallhl'ogrammon\\
dipl'asi'on >esti to~u ABG
trig'wnou;\\
<h g`ar >AG di'ametroc a>ut`o d'iqa t'emnei%
\eix{\gr{t'emnw}!\gr{t'emnei} 3rd sg pres ind act};\\
<'wste%
\eix{\gr{<'wste}} t`o ABGD parallhl'ogrammon\\
\kai%
\eix{\gr{ka`i}} to~u EBG trig'wnou >est`i dipl'asion.
}
{
E\c sittir \gk{ABG} \"u\c cgeni\\
\gk{EBG} \"u\c cgenine;\\
\gar\ onunla hem ayn\i\
\gk{BG} taban\i na sahiptir,\\
hem ayn\i\
\gk{BG} ve \gk{AE} paralelerindedir.\lli\\
Ama \gk{ABGD} paralelkenar\i,\\
\gk{ABG} \"u\c cgeninin iki kat\i d\i r;\\
\gar\ \gk{AG} k\"o\c segeni onu ikiye b\"oler;\\
\hoste\ \gk{ABGD} paralelkenar\i,\\
\gk{EBG} \"u\c cgeninin de iki kat\i d\i r.
}

\parsen{
Therefore, if a parallelogram\\
have the same base as a triangle,\\
and be in the same parallels,\\
double is\\
the parallelogram of the triangle;\\
\myqed
}
{
>E`an >'ara\eix{\gr{>'ara}} parallhl'ogrammon\\
trig'wnw|\\
b'asin te%
\eix{\gr{te}} >'eqh|%
\eix{\gr{>'eqw}!\gr{>'eqh"|} 3rd sg pres subj act} t`hn a>ut`hn\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic >~h|,\\
dipl'asi'on >est'i\\
t`o parallhl'ogrammon to~u trig'wnou;\\
<'oper >'edei de~ixai.
}
{
\Ara,  e\u ger bir paralelkenar\\
bir \"u\c cgenle\\
hem ayn\i\ tabana sahipse,\\
hem ayn\i\ paralelerdeyse,\\
iki kat\i d\i r\\
paralelkenar, \"u\c cgenin;\\
\ozqed.
}
\begin{center}
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\end{center}
\end{proposition}

\begin{proposition}{42}%Proposition I.42

\parsen{
To the given triangle equal,\\
a parallelogram to construct\\
in the given rectilineal angle.
}
{
T~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} trig'wnw| >'ison\\
parallhl'ogrammon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}\\
{}>en t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'ammw|.
}
{
Verilmi\c s bir \"u\c cgene e\c sit\\
bir paralelkenar\i\ in\c sa etmek\\
verilmi\c s bir d\"uzkenar a\c c\i da.
}

\parsen{
Let be\\
the given triangle \gk{ABG},\\
and the given rectilineal angle, \gk D.
}
{
>'Estw\\
t`o m`en%
\eix{\gr{m'en}} doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} tr'igwnon t`o ABG,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc <h D; 
}
{
Olsun\\
verilmi\c s \"u\c cgen \gk{ABG},\\
ve verilmi\c s d\"uzkenar a\c c\i\ \gk D.
}

\parsen{
It is necessary then\\
to the triangle \gk{ABG} equal\\
a parallelogram to construct\\
in the rectilineal angle \gk D.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
t~w| ABG trig'wnw| >'ison\\
parallhl'ogrammon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}\\
{}>en t~h| D gwn'ia| e>ujugr'ammw|.
}
{
\Deidee\\
 \gk{ABG}  \"u\c cgenine e\c sit\\
bir paralelkenar in\c sa etmek\\
\gk D d\"uzkenar a\c c\i s\i nda.
}

\parsen{
Suppose \gk{BG} has been cut in two at \gk E,\\
and there has been joined \gk{AE},\\
and there has been constructed\\
on the \strgt\ \gk{EG},\\
and at the point \gk E on it,\\
to angle \gk D equal,\\
\gk{GEZ},\\
also, through \gk A, parallel to \gk{EG},\\
suppose \gk{AH} has been drawn,\\
and through \gk G, parallel to \gk{EZ},\\
suppose \gk{GH} has been drawn;\\
therefore a parallelogram is \gk{ZEGH}.
}
{
Tetm'hsjw%
\eix{\gr{t'emnw}!\gr{tetm'hsjw} 3rd sg perf imperat mp} <h BG d'iqa kat`a t`o E,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h AE,\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
pr`oc t~h| EG e>uje'ia|\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| E\\
t~h| D gwn'ia| >'ish\\
<h <up`o GEZ,\\
\kai%
\eix{\gr{ka`i}} di`a m`en%
\eix{\gr{m'en}} to~u A t~h| EG par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h AH,\\
di`a d`e to~u G t~h| EZ par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h GH;\\
parallhl'ogrammon >'ara\eix{\gr{>'ara}} >est`i t`o ZEGH.
}
{
\gk{BG}, \gk E'da ikiye b\"olm\"u\c s olsun,\\
ve \gk{AE} birle\c stirilmi\c s olsun,\\
ve in\c sa edilmi\c s olsun\\
 \gk{EG} do\u grusunda,\\
ve \"uzerindeki \gk E noktas\i nda,\\
\gk D a\c c\i s\i na e\c sit,\\
\gk{GEZ},\\
ayr\i ca, \gk A'dan, \gk{EG}'ya paralel,\\
 \gk{AH} ilerletilmi\c s olsun,\\
ve  \gk G'dan,  \gk{EZ}'ya paralel,\\
\gk{GH} ilerletilmi\c s olsun;\\
\ara\ \gk{ZEGH} bir paralelkenard\i r.
}

\parsen{
And since equal is \gk{BE} to \gk{EG},\\
equal is also\\
triangle \gk{ABE} to triangle \gk{AEG};\\
for they are on equal bases,\\
\gk{BE} and \gk{EG},\\
and in the same parallels,\\
\gk{BG} and \gk{AH};\\
double therefore is\\
triangle \gk{ABG} of triangle \gk{AEG}.\\
also is\\
parallelogram \gk{ZEGH}\\
double of triangle \gk{AEG};\\
for it has the same base as it,\\
and\\
is in the same parallels as it;\\
therefore is equal\\
the parallelogram \gk{ZEGH}\\
to the triangle \gk{ABG}.\\
And it has angle \gk{GEZ}\\
equal to the given \gk D.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in\\
<h BE t~h| EG,\\
{}>'ison >est`i \kai%
\eix{\gr{ka`i}}
t`o ABE tr'igwnon\\
t~w| AEG trig'wnw|;\\
{}>ep'i te%
\eix{\gr{te}} g`ar >'iswn b'ase'wn e>isi
t~wn BE, EG\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic parall'hloic
ta~ic BG, AH;\\
dipl'asion >'ara\eix{\gr{>'ara}} >est`i\\
t`o ABG tr'igwnon to~u AEG trig'wnou.\\
{}>'esti d`e \kai%
\eix{\gr{ka`i}}
t`o ZEGH parallhl'ogrammon\\
dipl'asion to~u AEG trig'wnou;\\
b'asin te%
\eix{\gr{te}} g`ar a>ut~w| t`hn a>ut`hn >'eqei%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}\lli\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic >estin a>ut~w| parall'hloic;\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i\\
t`o ZEGH parallhl'ogrammon\\
t~w| ABG trig'wnw|.\\
\kai%
\eix{\gr{ka`i}} >'eqei%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} t`hn <up`o GEZ gwn'ian\\
{}>'ishn t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} t~h| D.
}
{
Ve e\c sit oldu\u gundan\\
\gk{BE}, \gk{EG}'ya,\\
\gk{ABE}  \"u\c cgeni de e\c sittir\\
\gk{AEG} \"u\c cgenine;\\
\gar\ hem e\c sit \gk{BE} ve \gk{EG} tabanlar\i nda,\lli\\
hem ayn\i\ \gk{BG} ve \gk{AH} paralelerindedir;\lli\\
iki kat\i d\i r \ara\\
\gk{ABG} \"u\c cgeni, \gk{AEG} \"u\c cgeninin,\\
ayr\i ca \gk{ZEGH} paralelkenar\i\lli\\
\gk{AEG} \"u\c cgeninin iki kat\i d\i r;\\
\gar\ hem onunla ayn\i\ tabana sahiptir\\
hem onunla ayn\i\ paralellerdedir;\lli\\
\ara\ e\c sittir\\
\gk{ZEGH} paralelkenar\i\\
\gk{ABG} \"u\c cgenine.\\
Ve onun \gk{GEZ} a\c c\i s\i\\
verilmi\c s \gk D'ya e\c sittir.
}

\parsen{
Therefore, to the given triangle \gk{ABG}\\
equal,\\
a parallelogram has been constructed,\\
\gk{ZEGH},\\
in the angle \gk{GEZ},\\
which is equal to \gk D;\\
\myqef
}
{
T~w| >'ara\eix{\gr{>'ara}} doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} trig'wnw| t~w| ABG\\
{}>'ison\\
parallhl'o\-gram\-mon sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp}
t`o ZEGH\\
{}>en gwn'ia| t~h| <up`o GEZ,\\
<'htic >est`in >'ish t~h| D;\\
<'oper >'edei poi~hsai.
}
{
\Ara,  verilmi\c s \gk{ABG} \"u\c cgenine\\
e\c sit\\
bir \gk{ZEGH} paralelkenar in\c sa edilmi\c sti\\
\gk{GEZ} a\c s\i s\i nda,\\
\gk D a\c s\i s\i na e\c sit olan;\\
\ozqef.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.3792187)(4.2665625,1.3792187)
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\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
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\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{43}%Proposition I.43

\parsen{
Of any parallelogram,\\
of the parallelograms about the diameter,\\
the complements\\
are equal to one another.
}
{
Pant`oc parallhlogr'ammou\\
t~wn per`i t`hn di'ametron\\
parallhlogr'ammwn\\
t`a paraplhr'wmata\\
{}>'isa >all'hloic >est'in.
} 
{
Herhangi bir paralelkenar\i n\\
k\"o\c segeni etraf\i ndaki\\
paralelkenarlar\i n\\
t\"umleyenleri,\\
birbirine e\c sittir.
}

\parsen{
Let there be\\
a parallelogram \gk{ABGD},\\
and its diameter, \gk{AG},\\
and about \gk{AG}\\
let be parallelograms,\\
\gk{EJ} and \gk{ZH},\footnotemark\\
and the so-called\footnotemark\ complements,\\
\gk{BK} and \gk{KD}.
}
{
>'Estw\\
parallhl'ogrammon t`o ABGD,\\
di'ametroc d`e a>uto~u <h AG,\\
per`i d`e t`hn AG\\
parallhl'ogramma m`en%
\eix{\gr{m'en}} >'estw\\
t`a EJ, ZH,\\
t`a d`e leg'omena%
\eix{\gr{l'egw}!\gr{leg'omenoc} part pres mp} paraplhr'wmata\\
t`a BK, KD;
}
{
Olsun\\
paralelkenar \gk{ABGD},\\
ve onun k\"o\c segeni \gk{AG},\\
ve \gk{AG} etraf\i nda\\
paralelkenarlar,\\
\gk{EJ} ve \gk{ZH} olsun,\footnotemark\\
ve s\"ozde t\"umleyenleri,\\
\gk{BK} ile \gk{KD}.
}
\footnotetext{Yunancada \gk{EJ} paralelkenar\i, \gfs{t`o EJ parallel'ogrammon} veya k\i saca \gfs{t`o EJ} iken, \gk{EJ} \c cizgisi, \gfs{<h EJ gramm'h} veya \gfs{<h EJ} olur.  Fark, harf{}i tarif{}le g\"osterilir.}
\myfntext{Here Euclid can use two letters without qualification for a parallelogram, because they are not unqualified in the Greek: they take the neuter article, while a line takes the feminine article.}
\myfntext{This is Heath's translation.  The Greek does not require
  anything corresponding to `so-'.  The LSJ lexicon \cite{LSJ} gives
  the present proposition as the original geometrical use of
  \gr{parapl'hrwma}---other meanings are `expletive' and a certain
  flowering herb.}

\parsen{
I say that\\
equal is the complement \gk{BK}\\
to the complement \gk{KD}.
}
{
l'egw, <'oti\\
{}>'ison >est`i t`o BK parapl'hrwma\\
t~w| KD paraplhr'wmati.
}
{
\Legohoti\\
\gk{BK} t\"umleyeni e\c sittir\\
\gk{KD} t\"umleyenine.
}


\parsen{
For, since a parallelogram is\\
\gk{ABGD},\\
and its diameter, \gk{AG},\\
equal is\\
triangle \gk{ABG} to triangle \gk{AGD}.\\
Moreover, since a parallelogram is\\
\gk{EJ},\\
and its diameter, \gk{AK},\\
equal is\\
triangle \gk{AEK} to triangle \gk{AJK}.\\
Then for the same [reasons] also\\
triangle \gk{KZG} to \gk{KHG} is equal.\\
Since then triangle \gk{AEK}\\
is equal to triangle \gk{AJK},\\
and \gk{KZG} to \gk{KHG},\\
triangle \gk{AEK} with \gk{KHG}\\
is equal\\
to triangle \gk{AJK} with \gk{KZG};\\
also is triangle \gk{ABG}, as a whole,\\
equal to \gk{ADG}, as a whole;\\
therefore the complement \gk{BK} remaining\\
to the complement \gk{KD} remaining\\
is equal.
}
{
>Epe`i%
\eix{\gr{>epe'i}} g`ar parallhl'ogramm'on >esti\\
t`o ABGD,\\
di'ametroc d`e a>uto~u <h AG,\\
{}>'ison >est`i\\
t`o ABG tr'igwnon t~w| AGD trig'wnw|.\\
p'alin\eix{\gr{p'alin}}, >epe`i%
\eix{\gr{>epe'i}} parallhl'ogramm'on >esti\\
t`o EJ,\\
di'ametroc d`e a>uto~u >estin <h AK,\\
{}>'ison >est`i\\
t`o AEK tr'igwnon t~w| AJK trig'wnw|.\\
di`a\eix{\gr{di'a}} t`a a>ut`a d`h%
\eix{\gr{d'h}} \kai%
\eix{\gr{ka`i}}\\
t`o KZG tr'igwnon t~w| KHG >estin >'ison.\\
{}>epe`i%
\eix{\gr{>epe'i}} o>~un\eix{\gr{o>~un}} t`o m`en%
\eix{\gr{m'en}} AEK tr'igwnon\\
t~w| AJK trig'wnw| >est`in >'ison,\\
t`o d`e KZG t~w| KHG,\\
t`o AEK tr'igwnon met`a to~u KHG\\
{}>'ison >est`i\\
t~w| AJK  trig'wnw| met`a to~u KZG;\\
{}>'esti d`e \kai%
\eix{\gr{ka`i}} <'olon t`o ABG tr'igwnon\\
<'olw| t~w| ADG >'ison;\\
loip`on >'ara\eix{\gr{>'ara}} t`o BK parapl'hrwma\\
loip~w| t~w| KD paraplhr'wmat'i\\
{}>estin >'ison.
}
{
\Gar\ bir paralelkenar oldu\u gundan\\
\gk{ABGD},\\
ve \gk{AG}, onun k\"o\c segeni [oldu\u gundan],\\
e\c sittir\\
\gk{ABG} \"u\c cgeni, \gk{AGD} \"u\c cgenine.\\
\Palin, bir paralelkenar oldu\u gundan\\
\gk{EJ},\\
ve \gk{AK}, onun k\"o\c segeni [oldu\u gundan],\\
e\c sittir\\
 \gk{AEK} \"u\c cgeni, \gk{AJK} \"u\c cgenine.\\
\Dee\ \diatauta\\
\gk{KZG} \"u\c cgeni de, \gk{KHG}'ya e\c sittir.\\
\Oun\ \gk{AEK} \"u\c cgeni,\\
\gk{AJK} \"u\c cgenine e\c sit oldu\u gundan,\\
ve \gk{KZG}, \gk{KHG}'ya,\\
 \gk{AEK} \"u\c cgeni, \gk{KHG} ile,\\
e\c sittir\\
\gk{AJK} \"u\c cgenine, \gk{KZG} ile;\\
ve b\"ut\"un \gk{ABG} \"u\c cgeni,\\
b\"ut\"un \gk{ADG}'ya e\c sittir;\\
\ara\ \kalan\ \gk{BK} t\"umleyeni,\\
\kalan\ \gk{KD} t\"umleyenine\\
e\c sittir.
}


\parsen{
Therefore, of any parallelogram area,\\
of the about-the-diameter\\
parallelograms,\\
the complements\\
are equal to one another;\\
\myqed
}
{
Pant`oc >'ara\eix{\gr{>'ara}} parallhlogr'ammou qwr'iou\\
t~wn per`i t`hn di'ametron\\
parallhlogr'ammwn\\
t`a paraplhr'wmata\\
{}>'isa >all'h\-loic >est'in;\\
<'oper >'edei de~ixai.
}
{
\Ara,  herhangi bir paralelkenar alan\i n\\
k\"o\c segeni etraf\i ndaki\\
paralelkenarlar\i n\\
t\"umleyenleri,\\
birbirine e\c sittir;\\
\ozqed.
}
\begin{center}
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}
\end{center}
\end{proposition}

\begin{proposition}{44}%Proposition I.44

\newparsen{
Par`a t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian\\
t~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass}  trig'wnw| >'ison\\
parallhl'ogrammon parabale~in%
\eix{\gr{parab'allw}!\gr{parabale~in} pres inf act}\\
{}>en  t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'am\-mw|.
}
{
Verilmi\c s bir do\u gru boyunca\\
verilmi\c s bir \"u\c cgene e\c sit,\\
bir paralelkenar uygulamak\\
verilmi\c s bir d\"uz kenar a\c c\i da.
}

\begin{center}
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}
\end{center}

\newparsen{
>'Estw\\
<h m`en%
\eix{\gr{m'en}} doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h AB,\\
t`o d`e doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} tr'igwnon t`o G,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc\verselinebreak <h D; 
}
{
Olsun\\
verilmi\c s do\u gru \gk{AB},\\
ve verilmi\c s \"u\c cgen \gk G,\\
ve verilmi\c s d\"uzkenar a\c c\i\verselinebreak\gk D.
}


\newparsen{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
par`a t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ian t`hn AB\\
t~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} trig'wnw| t~w| G >'ison\\
parallhl'ogrammon parabale~in%
\eix{\gr{parab'allw}!\gr{parabale~in} pres inf act}\\
{}>en >'ish| t~h| D gwn'ia|.
}
{
\Deidee\\
verilmi\c s \gk{AB} do\u grusu boyunca\\
verilmi\c s \gk G \"u\c cgenine e\c sit\\
bir paralelkenar\\
verilmi\c s \gk D a\c c\i s\i nda uygulamak.
}

\newparsen{
Sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
t~w| G trig'wnw| >'ison\\
parallhl'ogrammon t`o BEZH\\
{}>en gwn'ia| t~h| <up`o EBH,\\
<'h >estin >'ish t~h| D;\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
<'wste%
\eix{\gr{<'wste}} >ep> e>uje'iac e>~inai t`hn BE t~h| AB,\\
\kai%
\eix{\gr{ka`i}} di'hqjw%
\eix{\gr{>'agw}!\gr{di'agw}!\gr{di'hqjw} 3rd sg perf imperat mp}
<h ZH >ep`i t`o J,\\
\kai%
\eix{\gr{ka`i}} di`a to~u A\\
<opot'era| t~wn BH, EZ par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h AJ,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h JB. 
}
{
\.In\c sa edilmi\c s olsun\\
\gk G \"u\c cgenine e\c sit olan\\
\gk{BEZH} paralelkenar\i,\\
\gk{EBH} a\c c\i s\i nda,\\
\gk D'ya e\c sit olan;\\
ve oturtulmu\c s olsun\\
\hosteB\ \gk{BE}, \gk{AB} ile bir do\u gruda olsun,\\
ve \gk{ZH}, \gk J'a ilerletilmi\c s olsun\\
ve \gk A'dan,\\
\gk{BH} ve \gk{EZ}'dan birine paralel olan,\\
\gk{AJ} ilerletilmi\c s olsun,\\
ve \gk{JB} birle\c stirilmi\c s\ olsun.\\
}

\newparsen{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} e>ic parall'hlouc t`ac AJ, EZ\\
e>uje~ia >en'epesen%
\eix{\gr{p'iptw}!\gr{>emp'iptw}!\gr{>en'epese} 3rd sg aor ind act} 
<h JZ,\\
a<i >'ara\eix{\gr{>'ara}} <up`o  AJZ, JZE gwn'iai\\
dus`in >orja~ic e>isin >'isai.\\
a<i >'ara\eix{\gr{>'ara}} <up`o BJH, HZE\\
d'uo >orj~wn >el'asson'ec e>isin;\\
a<i d`e >ap`o >elass'onwn >`h d'uo >orj~wn\\
e>ic >'apeiron >ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}\\
sump'iptousin;\\
a<i JB, ZE >'ara\eix{\gr{>'ara}} >ekball'omenai%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekball'omenoc} part pres mp}\\
sumpeso~untai.
}
{
Ve  \gk{AJ} ile \gk{EZ} paralellerinin \"uzerine\\
\gk{JZ} do\u grusu d\"u\c st\"u\u g\"unden,\\
\gk{AJZ} ve \gk{JZE} a\c c\i lar\i\\
iki dik a\c c\i ya e\c sittir.\\
\Ara\ \gk{BJH} ve \gk{HZE}\\
iki dik a\c c\i dan k\"u\c c\"ukt\"ur.\\
Ve iki dik a\c c\i dan k\"u\c c\"uk olan,\\
sonsuza uzat\i lan,\\
\c carp\i\c s\i r.\\
\Ara\ uzat\i lan \gk{JB} ve \gk{ZE},\\
\c carp\i\c s\i r.
}

\newparsen{
>ekbebl'hsjwsan%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjwsan} 3rd pl perf imperat mp}\\
\kai%
\eix{\gr{ka`i}} sumpipt'etwsan kat`a t`o K,\\
\kai%
\eix{\gr{ka`i}} di`a to~u K shme'iou\\
<opot'era| t~wn EA, ZJ par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h KL,\\
\kai%
\eix{\gr{ka`i}} >ekbebl'hsjwsan%
\eix{\gr{b'allw}!\gr{>ekb'allw}!\gr{>ekbebl'hsjwsan} 3rd pl perf imperat mp} a<i JA, HB\\
{}>ep`i t`a L, M shme~ia.
}
{
Uzat\i lm\i\c s olsun\\
ve \gk K noktas\i nda \c carp\i\c sm\i\c s olsun,\\
ve \gk K noktas\i ndan,\\
\gk{EA} veya \gk{ZJ} do\u grusuna paralel olan,\\
\gk{KL} ilerletilmi\c s olsun,\\
ve \gk{JA} ve \gk{HB} uzat\i lm\i\c s olsun\\
\gk L ve \gk M'ye.
}

\newparsen{
parallhl'ogrammon >'ara\eix{\gr{>'ara}} >est`i t`o JLKZ,\\
di'ametroc d`e a>uto~u <h JK,\\
per`i d`e t`hn JK\\
parallhl'ogramma m`en%
\eix{\gr{m'en}} t`a AH, ME,\\
t`a d`e leg'omena%
\eix{\gr{l'egw}!\gr{leg'omenoc} part pres mp} 
paraplhr'wmata\\
t`a LB, BZ;\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i t`o LB t~w| BZ.\\
{}>all`a%
\eix{\gr{>all'a}} t`o BZ t~w| G trig'wnw| >est`in >'ison;\\
\kai%
\eix{\gr{ka`i}} t`o LB >'ara\eix{\gr{>'ara}} t~w| G >estin >'ison.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in\\
<h <up`o HBE gwn'ia t~h| <up`o ABM,\\
{}>all`a%
\eix{\gr{>all'a}} <h <up`o HBE t~h| D >estin >'ish,\\
\kai%
\eix{\gr{ka`i}} <h <up`o ABM >'ara\eix{\gr{>'ara}} t~h| D gwn'ia|\\
{}>est`in >'ish.
}
{
\Ara\ \gk{JLKZ} bir paralelkenard\i r,\lli\\
ve \gk{JK} onun k\"o\c segenidir,\\
ve  \gk{JK} etraf\i ndad\i r \\
 \gk{AH} ve \gk{ME} paralelkenarlar\i,\\
ve bunlar\i n s\"ozde t\"umleyenleri,\\
\gk{LB} ile \gk{BZ}'d\i r;\\
\Ara\ \gk{LB}, \gk{BZ}'ya e\c sittir.\\
Ama \gk{BZ}, \gk G \"u\c cgenine e\c sittir.\\
\Ara\ \gk{LB} da \gk G'ya e\c sittir.\\
Ve e\c sit oldu\u gundan\\
 \gk{HBE}  a\c c\i s\i,  \gk{ABM}'ye,\\
ama \gk{HBE}, \gk D'ya e\c sit oldu\u gundan,\\
\ara\ \gk{ABM} de \gk D a\c c\i s\i na\\
e\c sittir.
}

\newparsen{
Par`a t`hn doje~isan%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} >'ara\eix{\gr{>'ara}}
e>uje~ian t`hn AB\\
t~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} trig'wnw| t~w| G >'ison\\
parallhl'ogrammon parab'eblhtai%
\eix{\gr{parab'allw}!\gr{parab'eblhtai} 3rd sg perf mp} t`o LB\\
{}>en gwn'ia| t~h| <up`o ABM,\\
<'h >estin >'ish t~h| D;\\
<'oper >'edei poi~hsai.
}
{
\Ara, verilmi\c s \gk{AB} do\u grusu boyunca,\\
verilmi\c s bir \gk G \"u\c cgenine e\c sit olan,\\
\gk{LB} paralelkenar\i\ uygulanm\i\c s oldu,\lli\\
 \gk{ABM} a\c c\i s\i nda,\\
\gk D'ya e\c sit olan;\\
\ozqef.
}

\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.5592188)(8.543125,1.5592188)
\usefont{T1}{ptm}{m}{n}
\rput(4.3471875,-0.41421875){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(4.3465624,-1.4142188){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(5.222031,1.3857813){\gk E}
\usefont{T1}{ptm}{m}{n}
\rput(4.0226564,1.3457812){\gk Z}
\psline[linewidth=0.04cm](4.16,1.2007812)(8.16,1.2007812)
\psline[linewidth=0.04cm](8.16,1.2007812)(7.36,-1.1992188)
\psline[linewidth=0.04cm](7.34,-1.1992188)(3.36,-1.1792188)
\psline[linewidth=0.04cm](3.36,-1.1792188)(4.18,1.2207812)
\psline[linewidth=0.04cm](3.36,-1.1792188)(8.18,1.2007812)
\psline[linewidth=0.04cm](3.56,-0.5992187)(7.54,-0.57921875)
\psline[linewidth=0.04cm](5.16,1.1807812)(4.32,-1.1792188)
\usefont{T1}{ptm}{m}{n}
\rput(3.183125,-1.3942188){\gk J}
\usefont{T1}{ptm}{m}{n}
\rput(3.3453126,-0.6142188){\gk H}
\usefont{T1}{ptm}{m}{n}
\rput(8.368594,1.2857813){\gk K}
\usefont{T1}{ptm}{m}{n}
\rput(7.5625,-1.2942188){\gk L}
\usefont{T1}{ptm}{m}{n}
\rput(7.833281,-0.65421873){\gk M}
\usefont{T1}{ptm}{m}{n}
\rput(2.64,0.36578125){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(0.62109375,0.36578125){\gk G}
\psline[linewidth=0.04cm](2.72,1.2007812)(3.0,0.22078125)
\psline[linewidth=0.04cm](3.0,0.22078125)(2.22,-0.35921875)
\psline[linewidth=0.04cm](0.66,1.2207812)(1.48,0.0)
\psline[linewidth=0.04cm](1.48,0.0)(0.0,0.0)
\psline[linewidth=0.04cm](0.02,0.0)(0.68,1.2207812)
\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{45}%Proposition I.45

\parsen{
To the given rectilineal [figure] equal\\
a parallelogram to construct\\
in the given rectilineal angle.
}
{
T~w| doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>ujugr'ammw| >'ison\\
parallhl'ogrammon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}\\
{}>en t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| e>ujugr'ammw|.
}
{
Verilmi\c s bir d\"uzkenar [f{}ig\"ure] e\c sit\\
bir paralelkenar in\c sa etmek,\\
verilmi\c s d\"uzkenar a\c c\i da.
}

\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.5392188)(9.369375,1.5392188)
\usefont{T1}{ptm}{m}{n}
\rput(1.224375,-0.7342188){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(0.12375,1.0257813){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(3.1192188,0.00578125){\gk E}
\psline[linewidth=0.04cm](3.2371874,0.62078124)(2.6371875,-0.15921874)
\psline[linewidth=0.04cm](2.6371875,-0.15921874)(3.4971876,-0.29921874)
\usefont{T1}{ptm}{m}{n}
\rput(4.8998437,1.3057812){\gk Z}
\psline[linewidth=0.04cm](5.0571876,1.2007812)(9.057187,1.2007812)
\psline[linewidth=0.04cm](9.057187,1.2007812)(8.257188,-1.1992188)
\psline[linewidth=0.04cm](8.237187,-1.1992188)(4.2571874,-1.1792188)
\psline[linewidth=0.04cm](4.2571874,-1.1792188)(5.0771875,1.2207812)
\psline[linewidth=0.04cm](7.8571873,1.1807812)(6.9971876,-1.1792188)
\usefont{T1}{ptm}{m}{n}
\rput(6.9403124,-1.3942188){\gk J}
\usefont{T1}{ptm}{m}{n}
\rput(7.8825,1.3657813){\gk H}
\usefont{T1}{ptm}{m}{n}
\rput(4.105781,-1.3142188){\gk K}
\usefont{T1}{ptm}{m}{n}
\rput(9.219687,1.3057812){\gk L}
\usefont{T1}{ptm}{m}{n}
\rput(8.490469,-1.3542187){\gk M}
\usefont{T1}{ptm}{m}{n}
\rput(1.6771874,1.3257812){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(2.2982812,0.54578125){\gk G}
\psline[linewidth=0.04cm](1.4771875,1.2007812)(2.1771874,0.38078126)
\psline[linewidth=0.04cm](2.1771874,0.38078126)(1.3571875,-0.55921876)
\psline[linewidth=0.04cm](1.3571875,-0.55921876)(0.0771875,0.8407813)
\psline[linewidth=0.04cm](0.0971875,0.82078123)(1.4971875,1.1807812)
\psline[linewidth=0.04cm](1.4971875,1.1807812)(1.3571875,-0.5392187)
\end{pspicture} 
}
\end{center}

\parsen{
Let be\\
the given rectilineal [figure] \gk{ABGD},\\
and the given rectilineal angle, \gk E.
}
{
>'Estw\\
t`o m`en%
\eix{\gr{m'en}} doj`en%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uj'ugrammon t`o ABGD,\\
<h d`e doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia e>uj'ugrammoc <h E; 
}
{
Olsun\\
verilmi\c s d\"uzkenar [f{}ig\"ur] \gk{ABGD},\\
ve verilmi\c s d\"uzkenar a\c c\i\ \gk E.
}

\parsen{
It is necessary then\\
to the rectilineal \gk{ABGD} equal\\
a parallelogram to construct\\
in the given angle \gk E.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
t~w| ABGD e>uju\-gr'ammw| >'ison\\
parallhl'ogrammon sust'hsasjai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sust'hsasjai} aor inf mid causal}\\
{}>en t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} gwn'ia| t~h| E.
}
{
\Deidee\\
 \gk{ABGD} d\"uzkenar\i na e\c sit\\
bir paralelkenar in\c sa etmek,\\
verilmi\c s \gk E a\c c\i s\i nda.
}
\parsen{
Suppose has been joined \gk{DB},\\
and suppose has been constructed,\\
equal to the triangle \gk{ABD},\\
a parallelogram, \gk{ZJ},\\
in the angle \gk{JKZ},\\
which is equal to \gk E;\\
and suppose there has been applied\\
along the \strgt\ \gk{HJ},\\
equal to triangle \gk{DBG},\\
a parallelogram, \gk{HM},\\
in the angle \gk{HJM},\\
which is equal to \gk E.
}
{
>Epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h DB,\\
\kai%
\eix{\gr{ka`i}} sunest'atw%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sunest'atw} 3rd sg perf
  imperat mp}\\
t~w| ABD trig'wnw| >'ison\\
parallhl'ogrammon t`o ZJ\\
{}>en t~h| <up`o JKZ gwn'ia|,\\
<'h >estin >'ish t~h| E;\\
\kai%
\eix{\gr{ka`i}} parabebl'hsjw%
\eix{\gr{parab'allw}!\gr{parabebl'hsjw} 3rd sg perf imperat mp}\\
par`a t`hn HJ e>uje~ian\\
t~w| DBG trig'wnw| >'ison\\
parallhl'ogrammon t`o HM\\
{}>en t~h| <up`o HJM gwn'ia|,\\
<'h >estin >'ish t~h| E. 
}
{
\gk{DB} birle\c stirilmi\c s\ olsun,\\
ve in\c sa edilmi\c s\ olsun,\\
\gk{ABD} \"u\c cgenine e\c sit,\\
bir \gk{ZJ} paralelkenar\i,\\
\gk{JKZ} a\c c\i s\i nda,\\
\gk E'a e\c sit olan;\\
ve uygulanm\i\c s olsun\\
\gk{HJ} do\u grusu boyunca,\\
\gk{DBG} \"u\c cgenine e\c sit,\\
bir \gk{HM} paralelkenar\i,\\
\gk{HJM} a\c c\i s\i nda,\\
\gk E'a e\c sit olan.
}

\parsen{
And since angle \gk E\\
to either of \gk{JKZ} and \gk{HJM}\\
is equal,\\
therefore also \gk{JKZ} to \gk{HJM}\\
is equal.\\
Let \gk{KJH} be added in common;\\
therefore \gk{ZKJ} and \gk{KJH}\\
to \gk{KJH} and \gk{HJM}\\
are equal.\\
But \gk{ZKJ} and \gk{KJH}\\
are equal to two \rgt s;\\
therefore also \gk{KJH} and \gk{HJM}\\
are equal to two \rgt s.\\
Then to some \strgt, \gk{HJ},\\
and at the same point, \gk J,\\
two \strgt s, \gk{KJ} and \gk{JM},\\
not lying in the same parts,\\
the adjacent angles\\
make equal to two \rgt s.\\
In a \strgt\ then are \gk{KJ} and \gk{JM};\\
and since on the parallels \gk{KM} and \gk{ZH}\\
fell the \strgt\ \gk{JH},\\
the alternate angles \gk{MJH} and \gk{JHZ}\\
are equal to one another.\\
Let \gk{JHL} be added in common;\\
therefore \gk{MJH} and \gk{JHL}\\
to \gk{JHZ} and \gk{JHL}\\
are equal.\\
But \gk{MJH} and \gk{JHL}\\
are equal to two \rgt s;\\
therefore also \gk{JHZ} and \gk{JHL}\\
are equal to two \rgt s;\\
therefore on a \strgt\ are \gk{ZH} and \gk{HL}.\\
And since \gk{ZK} to \gk{JH}\\
is equal and parallel,\\
but also \gk{JH} to \gk{ML},\\
therefore also \gk{KZ} to \gk{ML}\\
is equal and parallel;\\
and join them\\
\gk{KM} and \gk{ZL}, which are \strgt s;\\
therefore also \gk{KM} and \gk{ZL}\\
are equal and parallel;\\
a parallelogram therefore is \gk{KZLM}.\\
And since equal is\\
triangle \gk{ABD}\\
to the parallelogram \gk{ZJ},\\
and \gk{DBG} to \gk{HM},\\
therefore, as a whole,\\
the rectilineal \gk{ABGD}\\
to parallelogram \gk{KZLM} as a whole\\
is equal.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} <h E gwn'ia\\
<ekat'era| t~wn <up`o JKZ, HJM\\
{}>estin >'ish,\\
\kai%
\eix{\gr{ka`i}} <h <up`o JKZ >'ara\eix{\gr{>'ara}}\\
t~h| <up`o HJM >estin >'ish.\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o KJH;\\
a<i >'ara\eix{\gr{>'ara}} <up`o ZKJ, KJH\\
ta~ic <up`o KJH, HJM >'isai e>is'in.\\
{}>all>%
\eix{\gr{>all'a}} a<i <up`o ZKJ, KJH\\ 
dus`in >orja~ic >'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o KJH, HJM >'ara\eix{\gr{>'ara}}\\
d'uo >orja~ic >'isai e>is'in.\\
pr`oc d'h tini e>uje~ia| t~h| HJ\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| J\\
d'uo e>uje~iai a<i KJ, JM\\ 
m`h >ep`i t`a a>ut`a m'erh ke'imenai\\
t`ac >efex~hc gwn'iac\\
d'uo >orja~ic >'isac poio~usin%
\eix{\gr{poi'ew}!\gr{poio~usin} 3rd pl pres ind act};\\
{}>ep> e>uje'iac >'ara\eix{\gr{>'ara}} >est`in <h KJ t~h| JM;\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} e>ic parall'hlouc t`ac KM, ZH\\
e>uje~ia >en'epesen <h JH,\\
a<i >enall`ax gwn'iai a<i <up`o MJH, JHZ\\
{}>'isai >all'hlaic e>is'in.\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o JHL;\\
a<i >'ara\eix{\gr{>'ara}} <up`o MJH, JHL\\
ta~ic <up`o JHZ, JHL >'isai e>isin.\\
{}>all>%
\eix{\gr{>all'a}} a<i <up`o MJH, JHL\\
d'uo >orja~ic >'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} a<i <up`o JHZ, JHL >'ara\eix{\gr{>'ara}}\\
d'uo >orja~ic >'isai e>is'in;\\
{}>ep> e>uje'iac >'ara\eix{\gr{>'ara}} >est`in <h ZH t~h| HL.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} <h ZK t~h| JH\\
{}>'ish te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhl'oc >estin,\\
{}>all`a%
\eix{\gr{>all'a}} \kai%
\eix{\gr{ka`i}} <h JH t~h| ML,\\
\kai%
\eix{\gr{ka`i}} <h KZ >'ara\eix{\gr{>'ara}} t~h| ML\\
{}>'ish te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhl'oc >estin;\\
\kai%
\eix{\gr{ka`i}} >epizeugn'uousin%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}} a>ut`ac e>uje~iai a<i KM, ZL;\\
\kai%
\eix{\gr{ka`i}} a<i KM, ZL >'ara\eix{\gr{>'ara}}\\
{}>'isai te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} par'allhlo'i e>isin;\\
parallhl'ogrammon >'ara\eix{\gr{>'ara}} >est`i t`o KZLM.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ison >est`i\\
t`o m`en%
\eix{\gr{m'en}} ABD tr'igwnon\\
t~w| ZJ parallhlogr'ammw|,\\
t`o d`e DBG t~w| HM,\\
<'olon >'ara\eix{\gr{>'ara}} t`o ABGD e>uj'ugrammon\lli\\
<'olw| t~w| KZLM parallhlogr'ammw|\\
{}>est`in >'ison.
}
{
Ve \gk E a\c c\i s\i\\
\gk{JKZ} ve \gk{HJM}'n\"un her birine\\
e\c sit oldu\u gundan,\\
\ara\ \gk{JKZ} da,\\
\gk{HJM}'ye e\c sittir.\\
Ortak olarak \gk{KJH} eklenmi\c s\ olsun;\\
\ara\ \gk{ZKJ} ve \gk{KJH},\\
\gk{KJH} ve \gk{HJM}'ye e\c sittir.\\
Ama \gk{ZKJ} ve \gk{KJH}\\
iki dik a\c c\i ya e\c sittir;\\
\ara\ \gk{KJH} ve \gk{HJM} de,\\
iki dik a\c c\i ya e\c sittir.\\
\Dee\ bir \gk{HJ} do\u grusuna,\\
ve ayn\i\ \gk J noktas\i nda,\\
iki \gk{KJ} ve \gk{JM} do\u grular\i,\\
ayn\i\ tarafta oturmayan,\\
biti\c sik a\c c\i lar\i\\
iki dik a\c c\i ya e\c sit yapar.\\
\Ara\ \gk{KJ}, \gk{JM} ile bir do\u grudad\i r;\\
ve \gk{KM} ve \gk{ZH} paralelleri \"uzerine\\
\gk{JH} do\u grusu d\"u\c st\"u\u g\"unden,\\
ters \gk{MJH} ve \gk{JHZ} a\c c\i lar\i\\
birbirine e\c sittir.\\
Ortak olarak \gk{JHL} eklenmi\c s olsun;\\
\ara\ \gk{MJH} ve \gk{JHL},\\
 \gk{JHZ} ve \gk{JHL}'ya e\c sittir.\\
Ama \gk{MJH} ve \gk{JHL}\\
iki dik a\c c\i ya e\c sittir;\\
\ara\ \gk{JHZ} ve \gk{JHL} da\\
iki dik a\c c\i ya e\c sittir;\\
\ara\ \gk{ZH}, \gk{HL} ile bir do\u grudad\i r.\\
Ve \gk{ZK}, \gk{JH}'ya\\
hem e\c sit hem paralel oldu\u gundan,\\
ama \gk{JH} da, \gk{ML}'ya,\\
\ara\ \gk{KZ} da \gk{ML}'ya\\
hem e\c sit hem paraleldir;\\
ve \gk{KM} ile \gk{ZL} do\u grular\i, onlar\i\ birle\c stirir;\\
\ara\ \gk{KM} ve \gk{ZL} da\\
hem e\c sit hem paraleldirler;\\
\ara\ \gk{KZLM} bir paralelkenard\i r.\lli\\
Ve e\c sit oldu\u gundan\\
 \gk{ABD} \"u\c cgeni\\
 \gk{ZJ} paralelkenar\i na,\\
ve \gk{DBG}, \gk{HM}'ye,\\
\ara, b\"ut\"un \gk{ABGD} d\"uzkenar [f{}ig\"ur\"u],\\
b\"ut\"un \gk{KZLM} paralelkenar\i na\\
e\c sittir.
}

\parsen{
Therefore, to the given rectilineal [figure], \gk{ABGD}, equal,\\
a parallelogram has been constructed,\\
\gk{KZLM},\\
in the angle \gk{ZKM},\\
which is equal to the given \gk E;\\
\myqef
}
{
T~w| >'ara\eix{\gr{>'ara}} doj'enti%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>ujugr'ammw| t~w| ABGD >'ison\\
parallhl'ogrammon sun'estatai%
\eix{\gr{<'isthmi}!\gr{sun'isthmi}!\gr{sun'estatai} 3rd sg perf ind mp}
t`o KZLM\\
{}>en gwn'ia| t~h| <up`o ZKM,\\
<'h >estin >'ish t~h| doje'ish|%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} t~h| E;\\
<'oper >'edei poi~hsai.
}
{
\Ara, verilmi\c s d\"uzkenar \gk{ABGD} f{}ig\"ur\"une e\c sit,\\
bir \gk{KZLM} paralelkenar\i\ in\c sa edilmi\c s\ oldu,\\
\gk{ZKM} a\c c\i s\i nda,\\
e\c sit olan verilmi\c s\ \gk E a\c c\i s\i na;\\
\ozqef.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.5392188)(9.369375,1.5392188)
\usefont{T1}{ptm}{m}{n}
\rput(1.224375,-0.7342188){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(0.12375,1.0257813){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(3.1192188,0.00578125){\gk E}
\psline[linewidth=0.04cm](3.2371874,0.62078124)(2.6371875,-0.15921874)
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\usefont{T1}{ptm}{m}{n}
\rput(4.8998437,1.3057812){\gk Z}
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\psline[linewidth=0.04cm](9.057187,1.2007812)(8.257188,-1.1992188)
\psline[linewidth=0.04cm](8.237187,-1.1992188)(4.2571874,-1.1792188)
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\usefont{T1}{ptm}{m}{n}
\rput(6.9403124,-1.3942188){\gk J}
\usefont{T1}{ptm}{m}{n}
\rput(7.8825,1.3657813){\gk H}
\usefont{T1}{ptm}{m}{n}
\rput(4.105781,-1.3142188){\gk K}
\usefont{T1}{ptm}{m}{n}
\rput(9.219687,1.3057812){\gk L}
\usefont{T1}{ptm}{m}{n}
\rput(8.490469,-1.3542187){\gk M}
\usefont{T1}{ptm}{m}{n}
\rput(1.6771874,1.3257812){\gk D}
\usefont{T1}{ptm}{m}{n}
\rput(2.2982812,0.54578125){\gk G}
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\psline[linewidth=0.04cm](1.4971875,1.1807812)(1.3571875,-0.5392187)
\end{pspicture} 
}
\end{center}

\end{proposition}



\begin{proposition}{46}%Proposition I.46

\parsen{
On the given \strgt\\
to set up a square.
}
{
>Ap`o t~hc doje'ishc%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje'iac\\
tetr'agwnon >anagr'ayai%
\eix{\gr{gr'afw}!\gr{>anagr'afw}!\gr{>anagr'ayai} aor inf act}.
}
{
Verilmi\c s bir do\u gruda\\
bir kare \c cizmak.
}
\parsen{
Let be\\
the given \strgt\ \gk{AB}.
}
{
>'Estw\\
<h doje~isa%
\eix{\gr{d'idwmi}!\gr{doje'ic} part aor pass} e>uje~ia <h AB;
}
{
Olsun\\
verilmi\c s do\u gru \gk{AB}.
}

\parsen{
It is required then\\
on the \strgt\ \gk{AB}\\
to set up a square.
}
{
de~i%
\eix{\gr{de~i}} d`h%
\eix{\gr{d'h}}\\
{}>ap`o t~hc AB e>uje'iac\\
tetr'agwnon >anagr'ayai%
\eix{\gr{gr'afw}!\gr{>anagr'afw}!\gr{>anagr'ayai} aor inf act}.
}
{
\Deidee\\
 \gk{AB} do\u grusunda\\
bir kare \c cizmek.
}


\parsen{
Suppose there has been drawn\\
to the \strgt\ \gk{AB},\\
at the point \gk A of it,\\
at a \rgt,\\
\gk{AG},\\
and suppose there has been laid down,\\
equal to \gk{AB},\\
\gk{AD};\\
and through the point \gk D,\\
parallel to \gk{AB},\\
suppose there has been drawn \gk{DE};\\
and through the point \gk B,\\
parallel to \gk{AD},\\
suppose there has been drawn \gk{BE}.\\
}
{
>'Hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp}\\
t~h| AB e>uje'ia|\\
{}>ap`o to~u pr`oc a>ut~h| shme'iou to~u A\\
pr`oc >orj`ac\\
<h AG,\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| AB >'ish\\
<h AD;\\
\kai%
\eix{\gr{ka`i}} di`a m`en%
\eix{\gr{m'en}} to~u D shme'iou\\
t~h| AB par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h DE,\\
di`a d`e to~u B shme'iou\\
t~h| AD par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h BE.
}
{
\.Ilerletilmi\c s olsun\\
\gk{AB} do\u grusunda,\\
onundaki \gk A noktas\i nda,\\
dik a\c c\i da,\\
\gk{AG},\\
ve oturmu\c s olsun,\\
\gk{AB}'ya e\c sit,\\
\gk{AD};\\
ve \gk D noktas\i ndan,\\
\gk{AB}'ya paralel,\\
\gk{DE} ilerletilmi\c s olsun;\\
ve \gk B noktas\i ndan,\\
 \gk{AD}'ya paralel,\\
\gk{BE} ilerletilmi\c s olsun.\\
}

\parsen{
A parallelogram therefore is \gk{ADEB};\\
equal therefore is \gk{AB} to \gk{DE},\\
and \gk{AD} to \gk{BE}.\\
But \gk{AB} to \gk{AD} is equal.\\
Therefore the four\\
\gk{BA}, \gk{AD}, \gk{DE}, and \gk{EB}\\
are equal to one another;\\
equilateral therefore\\
is the parallelogram \gk{ADEB}.
}
{
parallhl'ogrammon >'ara\eix{\gr{>'ara}} >est`i t`o ADEB;\\
{}>'ish >'ara\eix{\gr{>'ara}} >est`in <h m`en%
\eix{\gr{m'en}} AB t~h| DE,\\
<h d`e AD t~h| BE.\\
{}>all`a%
\eix{\gr{>all'a}} <h AB t~h| AD >estin >'ish;\\
a<i t'essarec >'ara\eix{\gr{>'ara}}\\
a<i BA, AD, DE, EB\\
{}>'isai >all'hlaic e>is'in;\\
{}>is'opleuron >'ara\eix{\gr{>'ara}}\\
{}>est`i t`o ADEB parallhl'ogrammon. 
}
{
\Ara\ \gk{ADEB} bir paralelkenard\i r;\lli\\
\ara\ \gk{AB}, \gk{DE}'a e\c sittir,\\
ve \gk{AD}, \gk{BE}'a.\\
Ama \gk{AB}, \gk{AD}'ya e\c sittir.\\
\Ara\ d\"ort\\
\gk{BA}, \gk{AD}, \gk{DE}, ve \gk{EB},\\
birbirine e\c sittir;\\
\ara\ e\c skenard\i r\\
 \gk{ADEB} paralelkenar\i.
}

\parsen{
I say then that\\
it is also right-angled.\\
}
{
l'egw d'h, <'oti\\
\kai%
\eix{\gr{ka`i}} >orjog'wnion.
}
{
\Dee\ \legohoti\\
dik a\c c\i l\i d\i r da.\\
}

\parsen{
For, since on the parallels \gk{AB} and \gk{DE}\\
fell the \strgt\ \gk{AD},\\
therefore the angles \gk{BAD} and \gk{ADE}\\
are equal to two \rgt s.\\
And \gk{BAD} is right;\\
right therefore is \gk{ADE}.\\
And of parallelogram areas\\
the opposite sides and angles\\
are equal to one another.\\
Right therefore is either\\
of the opposite angles \gk{ABE} and \gk{BED};\\
right-angled therefore is \gk{ADEB}.\\
And it was shown also equilateral.
}
{
>epe`i%
\eix{\gr{>epe'i}} g`ar e>ic parall'hlouc t`ac AB, DE\\
e>uje~ia >en'epesen <h AD,\\
a<i >'ara\eix{\gr{>'ara}} <up`o BAD, ADE gwn'iai\\
d'uo >orja~ic >'isai e>is'in.\\
{}>orj`h d`e <h <up`o BAD;\\
{}>orj`h >'ara\eix{\gr{>'ara}} \kai%
\eix{\gr{ka`i}} <h <up`o ADE.\\
t~wn d`e parallhlogr'ammwn qwr'iwn\\
a<i >apenant'ion pleura'i te%
\eix{\gr{te}} \kai%
\eix{\gr{ka`i}} gwn'iai\\
{}>'isai >all'hlaic e>is'in;\\
{}>orj`h >'ara\eix{\gr{>'ara}} \kai%
\eix{\gr{ka`i}} <ekat'era\\
t~wn >apenant'ion t~wn <up`o ABE, BED gwni~wn;\\
{}>orjog'wnion >'ara\eix{\gr{>'ara}} >est`i t`o ADEB.\\
{}>ede'iqjh\eix{\gr{de'iknumi}!\gr{>ede'iqjh} 3rd sg aor pass} d`e \kai%
\eix{\gr{ka`i}} >is'opleuron.
}
{
\Gar\  \gk{AB} ve \gk{DE} paralellerinin \"uzerine\\
\gk{AD} do\u grusu d\"u\c st\"u\u g\"unden,\\
\ara\ \gk{BAD} ve \gk{ADE},\\
iki dik a\c c\i ya e\c sittir.\\
Ve \gk{BAD} diktir;\\
\ara\ \gk{ADE} de diktir.\\
Ve paralelkenar alanlar\i n\\
hem kar\c s\i t kenar hem a\c c\i lar\i\\
birbirine e\c sittir.\\
\Ara\ diktir her biri\\
kar\c s\i t \gk{ABE} ve \gk{BED} a\c c\i lar\i ndan;\lli\\
\ara\ \gk{ADEB} dik a\c c\i l\i d\i r.\\
Ve g\"osterilmi\c sti ki e\c skenard\i r da.
}

\parsen{
A square therefore it is;\\
and it is on the \strgt\ \gk{AB}\\
set up;\\
\myqef
}
{
Tetr'agwnon >'ara\eix{\gr{>'ara}} >est'in;\\
ka'i >estin >ap`o t~hc AB e>uje'iac\\
{}>anagegramm'enon%
\eix{\gr{gr'afw}!\gr{>anagr'afw}!\gr{>anagegramm'enos} part perf mp};\\
<'oper >'edei poi~hsai.
}
{
\Ara\ bir karedir;\\
ve o \gk{AB} do\u grusu \"uzerine\\
\c cizilmi\c stir;\\
\ozqef.
}
\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.871875)(2.88875,1.871875)
\usefont{T1}{ptm}{m}{n}
\rput(2.724375,-1.726875){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(0.12375,-1.706875){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(2.7392187,0.493125){\gk E}
\usefont{T1}{ptm}{m}{n}
\rput(0.1571875,0.413125){\gk D}
\usefont{T1}{ptm}{m}{n}
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\psline[linewidth=0.04cm](2.5771875,0.428125)(0.3771875,0.428125)
\end{pspicture} 
}
\end{center}
\end{proposition}

\begin{proposition}{47}%Proposition I.47

\parsen{
In right-angled triangles,\\
the square on the side that subtends the right angle\\
is equal\\
to the squares on the sides that contain the right angle.
}
{
>En to~ic >orjogwn'ioic trig'wnoic\\
t`o >ap`o t~hc t`hn >orj`hn gwn'ian\\
<upoteino'ushc\\
pleur~ac tetr'agwnon\\
{}>'ison >est`i\\
to~ic >ap`o t~wn t`hn >orj`hn gwn'ian\\
perieqous~wn\\
pleur~wn tetrag'wnoic.
}
{
Dik a\c c\i l\i\ \"u\c cgenlerde,\\
dik a\c c\i y\i\\
rapteden\\
kenar\i n \"uzerindeki kare\\
e\c sittir\\
dik a\c c\i y\i\\
i\c ceren\\
kenarlar\i n \"uzerindeki karelere.
}


\begin{center}
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-2.6892188)(5.2784376,2.6892188)
\usefont{T1}{ptm}{m}{n}
\rput(1.0425,-0.36421874){\gk B}
\usefont{T1}{ptm}{m}{n}
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\usefont{T1}{ptm}{m}{n}
\rput(3.8373437,-2.5442188){\gk E}
\usefont{T1}{ptm}{m}{n}
\rput(1.1153125,-2.5042188){\gk D}
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\psline[linewidth=0.04cm](2.0753126,0.81078124)(2.0553124,-2.3692188)
\usefont{T1}{ptm}{m}{n}
\rput(0.09796875,0.39578125){\gk Z}
\usefont{T1}{ptm}{m}{n}
\rput(1.0584375,1.6157813){\gk H}
\usefont{T1}{ptm}{m}{n}
\rput(3.260625,2.5157812){\gk J}
\usefont{T1}{ptm}{m}{n}
\rput(5.103906,1.3157812){\gk K}
\end{pspicture} 
}
\end{center}

\parsen{
Let be\\
a right-angled triangle, \gk{ABG},\\
having the angle \gk{BAG} right.
}
{
>'Estw\\
tr'igwnon >orjog'wnion t`o ABG\\
{}>orj`hn >'eqon%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} t`hn <up`o BAG gwn'ian;
}
{
Olsun\\
dik a\c c\i l\i\ \"u\c cgen \gk{ABG},\\
dik a\c c\i s\i\ \gk{BAG} olan.
}


\parsen{
I say that\\
the square on \gk{GB}\\
is equal\\
to the squares on \gk{BA} and \gk{AG}.
}
{
l'egw, <'oti\\
t`o >ap`o t~hc BG tetr'agwnon\\
{}>'ison >est`i\\
to~ic >ap`o t~wn BA, AG tetrag'wnoic.
}
{
\Legohoti\\
\gk{GB} \"uzerindeki kare\\
e\c sittir\\
 \gk{BA} ve \gk{AG} \"uzerlerindeki karelere.
}

\parsen{
For, suppose there has been set up\\
on \gk{BG}\\
a square, \gk{BDEG},\\
and on \gk{BA} and \gk{AG},\\
\gk{HB} and \gk{JG},\\
and through \gk A,\\
parallel to either of \gk{BD} and \gk{GE},\\
suppose \gk{AL} has been drawn;\\
and suppose have been joined\\
\gk{AD} and \gk{ZG}.
}
{
>Anagegr'afjw%
\eix{\gr{gr'afw}!\gr{>anagr'afw}!\gr{>anagr'afjw} 3rd sg perf
  imperat mp} g`ar\\ 
{}>ap`o m`en%
\eix{\gr{m'en}} t~hc BG\\
tetr'agwnon t`o BDEG,\\
{}>ap`o d`e t~wn BA, AG\\
t`a HB, JG,\\
\kai%
\eix{\gr{ka`i}} di`a to~u A\\
<opot'era| t~wn BD, GE par'allhloc\\
{}>'hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} <h AL;\footnotemark\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjwsan%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjwsan} 3rd pl
  perf imperat mp}\\
a<i AD, ZG. 
}
{
\Gar\ \c cizilmi\c s olsun\\
\gk{BG} \"uzerinde\\
\gk{BDEG} karesi,\\
ve \gk{BA} ile \gk{AG} \"uzerlerinde,\\
\gk{HB} ve \gk{JG},\\
ve \gk A noktas\i ndan,\\
\gk{BD} ve \gk{GE}'a paralel olan,\\
\gk{AL} ilerletilmi\c s olsun;\\
ve birle\c stirilmi\c s\ olsun\\
\gk{AD} ve \gk{ZG}.
}
\footnotetext{Heiberg'in metninde \cite[p.~110]{Euclid-Heiberg} \gk L harf{}inin yerine \gk D harf{}i konulmu\c stur.%
% Heiberg's text has \gk D for \gk L at this place and elsewhere (though not in the diagram).  Probably this is a compositor's mistake, owing to the similarity in appearance of the two letters, especially in the font used.
}

\parsen{
And since right is\\
either of the angles \gk{BAG} and \gk{BAH},\\
on some \strgt, \gk{BA},\\
to the point \gk A on it,\\
two \strgt s, \gk{AG} and \gk{AH},\\
not lying in the same parts,\\
the adjacent angles\\
make equal to two \rgt s;\\
on a \strgt\ therefore is \gk{GA} with \gk{AH}.\\
Then for the same [reason]\\
also \gk{BA} with \gk{AJ} is on a \strgt.\\
And since equal is\\
angle \gk{DBG} to angle \gk{ZBA};\\
for either is \rgt;\\
let \gk{ABG} be added in common;\\
therefore \gk{DBA} as a whole\\
to \gk{ZBG} as a whole\\
is equal.\\
And since equal is\\
\gk{DB} to \gk{BG},\\
and \gk{ZB} to \gk{BA},\\
the two \gk{DB} and \gk{BA}\\
to the two \gk{ZB} and \gk{BG}\footnotemark\\
are equal,\\
either to either;\\
and angle \gk{DBA}\\
to angle \gk{ZBG}\\
is equal;\\
therefore the base \gk{AL}\\
to the base \gk{ZG}\\
{}[is] equal,\\
and the triangle \gk{ABD}\\
to the triangle \gk{ZBG}\\
is equal;\\
and of the triangle \gk{ABD}\\
the parallelogram \gk{BL} is double;\\
for they have the same base, \gk{BL},\\
and are in the same parallels,\\
\gk{BD} and \gk{AL};\\
and of the triangle \gk{ZBG}\\
the square \gk{HB} is double;\\
for again they have the same base,\\
\gk{ZB},\\
and are in the same parallels,\\
\gk{ZB} and \gk{HG}.\\
{}[And of equals,\\
the doubles are equal to one another.]\\
Equal therefore is\\
also the parallelogram \gk{BL}\\
to the square \gk{HB}.\\
Similarly then,\\
there being joined \gk{AE} and \gk{BK},\\
it will be shown that\\
also the parallelogram \gk{GL}\\
{}[is] equal to the square \gk{JG}.\\
Therefore the square \gk{DBEG} as a whole\\
to the two squares \gk{HB} and \gk{JG}\\
is equal.\\
Also is\\
the square \gk{BDEG} set up on \gk{BG},\\
and \gk{HB} and \gk{JG} on \gk{BA} and \gk{AG}.\\
Therefore the square on the side \gk{BG}\\
is equal\\
to the squares on the sides \gk{BA} and \gk{AG}.
}
{
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >orj'h >estin\\
<ekat'era t~wn <up`o BAG, BAH gwni~wn,\\
pr`oc d'h tini e>uje'ia| t~h| BA\\
\kai%
\eix{\gr{ka`i}} t~w| pr`oc a>ut~h| shme'iw| t~w| A\\
d'uo e>uje~iai a<i AG, AH\\
m`h >ep`i t`a a>ut`a m'erh ke'imenai%
\eix{\gr{ke'imai}!\gr{ke'imenoc} part pres mp}\\
t`ac >efex~hc gwn'iac\\
dus`in >orja~ic >'isac poio~usin%
\eix{\gr{poi'ew}!\gr{poio~usin} 3rd pl pres ind act};\\
{}>ep> e>uje'iac >'ara\eix{\gr{>'ara}} >est`in <h GA t~h| AH.\\
di`a\eix{\gr{di'a}} t`a a>ut`a d`h%
\eix{\gr{d'h}}\\
\kai%
\eix{\gr{ka`i}} <h BA t~h| AJ >estin >ep> e>uje'iac.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in\\
<h <up`o DBG gwn'ia t~h| <up`o ZBA;\\
{}>orj`h g`ar <ekat'era;\\
koin`h proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp} <h <up`o ABG;\\
<'olh >'ara\eix{\gr{>'ara}} <h <up`o DBA\\
<'olh| t~h| <up`o ZBG >estin >'ish.\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in\\
<h m`en%
\eix{\gr{m'en}} DB t~h| BG,\\
<h d`e ZB t~h| BA,\\
d'uo d`h%
\eix{\gr{d'h}} a<i DB, BA\\
d'uo ta~ic ZB, BG >'isai  e>is`in\\
<ekat'era <ekat'era|;\\
\kai%
\eix{\gr{ka`i}} gwn'ia <h <up`o DBA\\
gwn'ia| t~h| <up`o ZBG >'ish;\\
b'asic >'ara\eix{\gr{>'ara}} <h AD\\
b'asei t~h| ZG [>estin] >'ish,\\
\kai%
\eix{\gr{ka`i}} t`o ABD tr'igwnon\\
t~w| ZBG trig'wnw| >est`in >'ison;\\
ka'i [>esti] to~u m`en%
\eix{\gr{m'en}} ABD trig'wnou\\
dipl'asion t`o BL parallhl'ogrammon;\\
b'asin te%
\eix{\gr{te}} g`ar t`hn a>ut`hn >'eqousi%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act} t`hn BD\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic e>isi parall'hloic ta~ic BD, AL;\\
to~u d`e ZBG trig'wnou\\
dipl'asion t`o HB tetr'agwnon;\\
b'asin te%
\eix{\gr{te}} g`ar p'alin\eix{\gr{p'alin}} t`hn a>ut`hn >'eqousi%
\eix{\gr{>'eqw}!\gr{>'eqwn} part pres act}
t`hn ZB\\
\kai%
\eix{\gr{ka`i}} >en ta~ic a>uta~ic e>isi parall'hloic
ta~ic ZB, HG.\\
{}[t`a d`e t~wn >'iswn\\
dipl'asia >'isa >all'hloic >est'in;]\\
{}>'ison >'ara\eix{\gr{>'ara}} >est`i\\
\kai%
\eix{\gr{ka`i}} t`o BL parallhl'ogrammon\\
t~w| HB tetrag'wnw|.\\
<omo'iwc d`h%
\eix{\gr{d'h}}\\
{}>epizeugnum'enwn%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epizeugn'umenoc} part pres mp} t~wn AE, BK\\
deiqj'hsetai%
\eix{\gr{de'iknumi}!\gr{deiqj'hsetai} 3rd sg fut ind mp}\\
\kai%
\eix{\gr{ka`i}} t`o GL parallhl'ogrammon\\
{}>'ison t~w| JG tetrag'wnw|;\\
<'olon >'ara\eix{\gr{>'ara}} t`o BDEG tetr'agwnon\\
dus`i to~ic HB, JG tetrag'wnoic\\
{}>'ison >est'in.\\
ka'i >esti t`o m`en%
\eix{\gr{m'en}} BDEG tetr'agwnon\\
{}>ap`o t~hc BG >anagraf'en%
\eix{\gr{gr'afw}!\gr{>anagr'afw}!\gr{>anagr'ayai} aor inf act},\\
t`a d`e HB, JG >ap`o t~wn BA, AG.\\
t`o >'ara\eix{\gr{>'ara}} >ap`o t~hc BG pleur~ac tetr'agwnon\\
{}>'ison >est`i\\
to~ic >ap`o t~wn BA, AG pleur~wn tetrag'wnoic.
}
{
Ve dik oldu\u gundan\\
 \gk{BAG} ve \gk{BAH} a\c c\i lar\i n\i n her biri,\\
bir \gk{BA} do\u grusunda,\\
ve \"uzerindeki \gk A noktas\i nda,\\
\gk{AG} ve \gk{AH} do\u grular\i,\\
ayn\i\ tarafta oturmayan,\\
biti\c sik a\c c\i lar\\
iki dik a\c c\i ya e\c sit yapar;\\
\ara\ \gk{GA}, \gk{AH} ile bir do\u grudad\i r.\\
\Dee\ \diatauta\\
\gk{BA} da \gk{AJ} ile bir do\u grudad\i r.\\
Ve e\c sit oldu\u gundan\\
\gk{DBG} a\c c\i s\i, \gk{ZBA}'ya,\\
\gar\ her ikiside diktir;\\
ortak olarak \gk{ABG} eklenmi\c s\ olsun;\\
\ara\ b\"ut\"un \gk{DBA},\\
b\"ut\"un \gk{ZBG}'ya e\c sittir.\\
Ve e\c sit oldu\u gundan\\
\gk{DB}, \gk{BG}'ya,\\
ve \gk{ZB}, \gk{BA}'ya,\\
\dee\ \gk{DB} ve  \gk{BA} ikilisi\\
\gk{ZB} ve \gk{BG} ikilisine
e\c sittir,\\
her biri birine;\\
ve \gk{DBA} a\c c\i s\i\\
\gk{ZBG} a\c c\i s\i na e\c sittir;\\
\ara\ \gk{AL} taban\i\\
\gk{ZG} taban\i na e\c sittir,\\
ve \gk{ABD} \"u\c cgeni\\
\gk{ZBG} \"u\c cgenine e\c sittir;\\
ve \gk{ABD} \"u\c cgeninin\\
\gk{BL} paralelkenar\i n\i n iki kat\i d\i r;\\
\gar\ hem ayn\i\ \gk{BL} taban\i na sahiptir,\lli\\
hem ayn\i\ \gk{BD} ve \gk{AL} parallerindedir;\lli\\
ve \gk{ZBG} \"u\c cgeninin\\
\gk{HB} karesinin iki kat\i d\i r;\\
\gar\ \palin\ hem ayn\i\ \gk{ZB} taban\i na sahiptir\\
hem ayn\i\ \gk{ZB} ve \gk{HG} parallerindedir.\lli\\
{}[Ve e\c sitlerin\\
iki katlar\i\ birbirine e\c sittir.]\\
\Ara\ e\c sittir\\
\gk{BL} paralelkenar\i\ da\\
\gk{HB} karesine.\\
\Dee\ benzer \c sekilde,\\
\gk{AE} ve \gk{BK} birle\c stirilince,\\
g\"osterilecek ki\\
\gk{GL} paralelkenar\i\ da\\
\gk{JG} karesine e\c sittir.\\
\Ara\ b\"ut\"un \gk{DBEG}\\
iki \gk{HB} ve \gk{JG} karelerine\\
e\c sittir.\\
Ve \gk{BDEG} karesi,\\
\gk{BG} \"uzerine \c cizilmi\c stir,\\
ve \gk{HB} ve \gk{JG}, \gk{BA} ve \gk{AG} \"uzerine.\\
\Ara\ \gk{BG} kenar\i ndaki kare\lli\\
e\c sittir\\
\gk{BA} ve \gk{AG} kenarlar\i ndaki karelere.
}
\myfntext{Fitzpatrick considers this ordering of the two straight lines to be `obviously a mistake'.  But if it is a mistake, how could it have been made?}

\parsen{
Therefore in right-angled triangles\\
the square on the side subtending the right angle\\
is equal\\
to the squares on the sides subtending the right [angle];\\
\myqed
}
{
>En >'ara\eix{\gr{>'ara}}  to~ic >orjogwn'ioic trig'wnoic\\
t`o >ap`o t~hc t`hn >orj`hn gwn'ian\\
<upoteino'ushc\\
pleur~ac tetr'agwnon\\
{}>'ison >est`i\\
to~ic >ap`o t~wn t`hn >orj`hn [gwn'ian]\\
perieqous~wn\\
pleur~wn tetrag'wnoic;\\
<'oper >'edei de~ixai.
}
{
\Ara\ dik a\c c\i l\i\ \"u\c cgenlerde,\\
dik a\c c\i\\
rasteden\\
kenar \"uzerindeki kare\\
e\c sittir\\
dik a\c c\i y\i\\
i\c ceren\\
kenarlar\i n \"uzerindekilere;\\
\ozqed.
}

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}
\end{center}
\end{proposition}

\begin{proposition}{48}%Proposition I.48

\parsen{
If of a triangle\\
the square on one of the sides\\
be equal\\
to the squares on the remaining sides of the triangle,\\
the angle contained\\
by the two remaining sides of the triangle\\
is right.
}
{
>E`an trig'wnou\\
t`o >ap`o mi~ac t~wn pleur~wn tetr'agwnon\\
{}>'ison >~h|\\
to~ic >ap`o t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn tetrag'wnoic,\\
<h perieqom'enh%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp} gwn'ia\\
<up`o t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn\\
{}>orj'h >estin.
}
{
E\u ger bir \"u\c cgenin\\
bir kenar\i n\i n \"uzerindeki kare\\
e\c sitse\\
\"u\c cgenin \kalan\\
iki kenar\i ndaki karelere,\\
i\c cerilen a\c c\i\\
\"u\c cgenin \kalan\\
iki kenar\i\ taraf\i ndan,\\ 
diktir.
}
\parsen{
For, of the triangle \gk{ABG}\\
the square on the one side \gk{BG}\\
---suppose it is equal\\
to the squares on the sides \gk{BA} and \gk{AG}.
}
{
Trig'wnou g`ar to~u ABG\\
t`o >ap`o mi~ac t~hc BG pleur~ac\\
tetr'agwnon >'ison >'estw\\
to~ic >ap`o t~wn BA, AG pleur~wn\\
tetrag'wnoic;
}
{
\Gar\  \gk{ABG} \"u\c cgeninin\\
\gk{BG} kenar\i ndaki\\
karesi e\c sit olsun\\
\gk{BA} ve \gk{AG} kenarlar\i ndaki\\
karelere.
}


\parsen{
I say that\\
right is the angle \gk{BAG}.
}
{
l'egw, <'oti\\
{}>orj'h >estin <h <up`o BAG gwn'ia.
}
{
\Legohoti\\
\gk{BAG} a\c c\i s\i\ diktir.
}


\parsen{
For, suppose has been drawn\\
from the point \gk A\\
to the \strgt\ \gk{AG}\\
at \rgt s\\
\gk{AD},\\
and let be laid down\\
equal to \gk{BA}\\
\gk{AD},\\
and suppose \gk{DG} has been joined.
}
{
>'Hqjw\eix{\gr{>'agw}!\gr{>'hqjw} 3rd sg perf imperat mp} g`ar\\
{}>ap`o to~u A shme'iou\\
t~h| AG e>uje'ia|\\
pr`oc >orj`ac <h AD\\
\kai%
\eix{\gr{ka`i}} ke'isjw%
\eix{\gr{ke~imai}!\gr{ke'isjw} 3rd sg pres imperat mp}\\
t~h| BA >'ish <h AD,\\
\kai%
\eix{\gr{ka`i}} >epeze'uqjw%
\eix{\gr{ze'ugnumi}!\gr{>epize'ugnumi}!\gr{>epeze'uqjw} 3rd sg
  perf imperat mp} <h DG.
}
{
\Gar\ ilerletilmi\c s olsun\\
\gk A noktas\i ndan\\
\gk{AG} do\u grusuna\\
dik a\c c\i larda \gk{AD},\\
ve oturmu\c s olsun\\
\gk{BA}'ya e\c sit \gk{AD},\\
ve \gk{DG} birle\c stirilmi\c s\ olsun.
}
\parsen{
Since equal is \gk{DA} to \gk{AB},\\
equal is\\
also the square on \gk{DA}\\
to the square on \gk{AB}.\\
Let be added in common\\
the square on \gk{AG};\\
therefore the squares on \gk{DA} and \gk{AG}\\
are equal\\
to the squares on \gk{BA} and \gk{AG}.\\
But those on \gk{DA} and \gk{AG}\\
are equal\\
to that on \gk{DG};\\
for right is the angle \gk{DAG};\\
and those on \gk{BA} and \gk{AG}\\
are equal\\
to that on \gk{BG};\\
for it is supposed;\\
therefore the square on \gk{DG}\\
is equal\\
to the square on \gk{BG};\\
so that the side \gk{DG}\\
to the side \gk{BG}\\
is equal;\\
and since equal is \gk{DA} to \gk{AB},\\
and common [is] \gk{AG},\\
the two \gk{DA} and \gk{AG}\\
to the two \gk{BA} and \gk{AG}\\
are equal;\\
and the base \gk{DA}\\
to the base \gk{BG}\\
{}[is] equal;\\
therefore the angle \gk{DAG}\\
to the angle \gk{BAG}\\
{}[is] equal.\\
And right [is] \gk{DAG};\\
right therefore [is] \gk{BAG}.
}
{
>epe`i%
\eix{\gr{>epe'i}} >'ish >est`in <h DA t~h| AB,\\
{}>'ison >est`i\\
\kai%
\eix{\gr{ka`i}} t`o >ap`o t~hc DA tetr'agwnon\\
t~w| >ap`o t~hc AB tetrag'wnw|.\\
koin`on proske'isjw%
\eix{\gr{ke~imai}!\gr{proske~imai}!\gr{proske'isjw} 3rd sg pres imperat mp}\\
t`o >ap`o t~hc AG tetr'agwnon;\\
t`a >'ara\eix{\gr{>'ara}} >ap`o t~wn DA, AG\\
tetr'agwna >'isa >est`i\\
to~ic >ap`o t~wn BA, AG tetrag'wnoic.\\
{}>all`a%
\eix{\gr{>all'a}} to~ic m`en%
\eix{\gr{m'en}} >ap`o t~wn DA, AG\\
{}>'ison >est`i\\
t`o >ap`o t~hc DG;\\
{}>orj`h g'ar\eix{\gr{g'ar}} >estin <h <up`o DAG gwn'ia;\\
to~ic d`e >ap`o t~wn BA, AG\\
{}>'ison >est`i\\
t`o >ap`o t~hc BG;\\
<up'okeitai%
\eix{\gr{ke~imai}!\gr{<up'okeimai}} g'ar\eix{\gr{g'ar}};\\
t`o >'ara\eix{\gr{>'ara}} >ap`o t~hc DG tetr'agwnon\\
{}>'ison >est`i\\
t~w| >ap`o t~hc BG tetrag'wnw|;\\
<'wste%
\eix{\gr{<'wste}} \kai%
\eix{\gr{ka`i}} pleur`a <h DG\\
t~h| BG >estin >'ish;\\
\kai%
\eix{\gr{ka`i}} >epe`i%
\eix{\gr{>epe'i}} >'ish >est`in <h DA t~h| AB,\\
koin`h d`e <h AG,\\
d'uo d`h%
\eix{\gr{d'h}} a<i DA, AG\\
d'uo ta~ic BA, AG >'isai e>is'in;\\
\kai%
\eix{\gr{ka`i}} b'asic <h DG\\
b'asei t~h| BG >'ish;\\
gwn'ia >'ara\eix{\gr{>'ara}} <h <up`o DAG\\
gwn'ia| t~h| <up`o BAG [>estin] >'ish.\\
{}>orj`h d`e <h <up`o DAG;\\
{}>orj`h >'ara\eix{\gr{>'ara}} \kai%
\eix{\gr{ka`i}} <h <up`o BAG.
}
{
\gk{DA}, \gk{AB}'ya e\c sit oldu\u gundan,\\
e\c sittir\\
 \gk{DA} \"uzerindeki kare de\\
 \gk{AB} \"uzerindeki kareye.\\
Eklenmi\c s\ olsun ortak\\
\gk{AG} \"uzerindeki kare;\\
\ara\ \gk{DA} ve \gk{AG} \"uzerlerindeki\\
kareler e\c sittir\\
\gk{BA}  ve \gk{AG} \"uzerlerindeki karelere.\\
Ama \gk{DA} ve \gk{AG} \"uzerlerindekilere\\
e\c sittir\\
\gk{DG} \"uzerindekine;\\
\gar\ \gk{DAG} a\c c\i s\i\ diktir;\\
ve \gk{BA} ile  \gk{AG} \"uzerlerindekilere de\\
e\c sittir\\
\gk{BG} \"uzerindeki;\\
\gar\ kabul edilir;\\
\ara\ \gk{DG} \"uzerindeki kare\\
e\c sittir\\
\gk{BG} \"uzerindeki kareye;\\
\hoste\ \gk{DG} kenar\i\ da\\
 \gk{BG} kenar\i na e\c sittir;\\
ve  \gk{DA}, \gk{AB}'ya e\c sit oldu\u gundan,\\
ve \gk{AG} ortak [oldu\u gundan],\\
\gk{DA} ve \gk{AG} ikilisi\\
\gk{BA} ve \gk{AG} ikilisine e\c sittir;\\
ve \gk{DA} taban\i\\
\gk{BG} taban\i na e\c sittir;\\
\ara\ \gk{DAG} a\c c\i s\i\\
 \gk{BAG} a\c c\i s\i na e\c sittir.\\
Ve  \gk{DAG} diktir;\\
\ara\ diktir \gk{BAG}.
}
\parsen{
If, therefore, of a triangle,\\
the square on one of the sides\\
be equal\\
to the squares on the remaining two sides,\\
the angle contained\\
by the remaining two sides of the triangle\\
is right;\\
\myqed
}
{
>E`an >ar`a trig'wnou\\
t`o >ap`o mi~ac t~wn pleur~wn tetr'agwnon\\
{}>'ison >~h|\\
to~ic >ap`o t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn tetrag'wnoic,\\
<h perieqom'enh%
\eix{\gr{>'eqw}!\gr{peri'eqw}!\gr{perieq'omenoc} part pres mp} gwn'ia\\
<up`o t~wn loip~wn to~u trig'wnou\\
d'uo pleur~wn\\
{}>orj'h >estin;\\
<'oper >'edei de~ixai.
}
{
E\u ger \ara\  bir \"u\c cgende\\
bir kenar\i n \"uzerindeki kare\\
e\c sitse\\
\"u\c cgenin \kalan\\
iki kenarlar\i ndaki karelere,\\
i\c cerilen a\c c\i\\
\"u\c cgenin \kalan\\
iki kenarlar\i\ taraf\i ndan,\\
diktir;\\
\ozqed.
}
\begin{center}
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\psline[linewidth=0.04cm](1.35375,1.218125)(0.35375,-1.161875)
\psline[linewidth=0.04cm](0.37375,-1.161875)(2.35375,-1.181875)
\psline[linewidth=0.04cm](2.35375,-1.181875)(1.35375,1.218125)
\psline[linewidth=0.04cm](1.35375,1.218125)(1.33375,-1.161875)
\usefont{T1}{ptm}{m}{n}
\rput(1.3003125,-1.376875){\gk A}
\usefont{T1}{ptm}{m}{n}
\rput(2.5409374,-1.276875){\gk B}
\usefont{T1}{ptm}{m}{n}
\rput(1.4748437,1.343125){\gk G}
\usefont{T1}{ptm}{m}{n}
\rput(0.11375,-1.236875){\gk D}
\end{pspicture} 
\end{center}

\end{proposition}

%\end{comment}

%\addtocontents{toc}{\protect\end{multicols}}

%\appendix

\addchap{F{}iiller S\"ozl\"u\u g\"u}

\begin{description}
\item[\gfs{>'agw}] ilerle=
  \begin{description}
  \item[\gfs{di'agw}] ilerlet=
  \end{description}
\item[\gfs{<air'ew}]
  \begin{description}
  \item[\gfs{>afair'ew}] ay\i r=
  \end{description}
\item[\gfs{>ait'ew}] rica et=
\item[\gfs{>all'attw}]\mbox{}
  \begin{description}
  \item[\gfs{parall'attw}] sap=
  \end{description}
\item[\gfs{<'aptw}] \emph{med.}\ dokun=
\item[\gfs{<arm'ozw}]\mbox{}
  \begin{description}
  \item[\gfs{>efarm'ozw}] uygula=
  \end{description}
\item[\gfs{b'allw}]\mbox{}
  \begin{description}
  \item[\gfs{>ekb'allw}] uzat=
  \item[\gfs{parab'allw}] uygula=
  \item[\gfs{prosekb'allw}] uzat=
  \end{description}
\item[\gfs{gr'afw}] \c ciz=
  \begin{description}
  \item[\gfs{>anagr'afw}] \c ciz=
  \end{description}
\item[\gfs{>'eqw}] -i ol=
  \begin{description}
  \item[\gfs{peri'eqw}] i\c cer=
  \end{description}
\item[\gfs{ze'ugnumi}] birle\c stir=
\item[\gfs{<'isthmi}] dik=
  \begin{description}
  \item[\gfs{di-'isthmi}] (\gfs{di'asthma} uzunluk)
  \item[\gfs{>ef'isthmi}] -in \"uzerine dik=
  \item[\gfs{sun'isthmi}] in\c sa et=
  \end{description}
\item[\gfs{kal'ew}] \emph{med.} -e den=
\item[\gfs{ke~imai}] otur=
  \begin{description}
  \item[\gfs{>ekke~imai}] oturtul=
  \item[\gfs{proske~imai}] eklen=
  \item[\gfs{<upoke~imai}] kabul edil=
  \end{description}
\item[\gfs{lamb'anw}] al=
  \begin{description}
  \item[\gfs{>apolamb'anw}] ay\i r=
  \end{description}
\item[\gfs{l'egw}] (\gfs{leg'omenoc} s\"ozde)
\item[\gfs{pera'inw}] s\i n\i rla=
\item[\gfs{perat'ow}] s\i n\i rland\i r=
\item[\gfs{p'iptw}]\mbox{}
  \begin{description}
  \item[\gfs{>emp'iptw}] \"uzerine d\"u\c s=
  \item[\gfs{prosp'iptw}] (\emph{acc.}\ ile) \"uzerine d\"u\c s=
  \item[\gfs{sump'iptw}] \c carp\i\c s=
  \end{description}
\item[\gfs{poi'ew}] yap=
\item[\gfs{te'inw}]\mbox{}
  \begin{description}
  \item[\gfs{<upote'inw}] raptet=
  \end{description}
\item[\gfs{t'emnw}] kes=
  \begin{description}
  \item[\gfs{d'iqa t'emnw}] ikiye b\"ol=
  \end{description}
\item[\gfs{t'ijhmi}] yerle\c stir=
\end{description}




\addchap{Edatlar S\"ozl\"u\u g\"u}

\begin{description}
\item[\gfs{>all'a}] ama
\item[\gfs{>'ara}] \ara
\item[\gfs{di'a}] \dia
\item[\gfs{di`a ta>ut'a, di`a t`a a>ut'a}] \diatauta
\item[\gfs{g'ar}] \gar
\item[{[\emph{genitivus absolutus}]}] -ince
\item[\gfs{d'h}] \dee
\item[\gfs{>epe'i}] -di\u ginden
\item[\gfs{ka'i}] de, ve
\item[\gfs{m'en\dots d'e}] ---
\item[\gfs{m'hn}] \meen
\item[\gfs{o>~un}] \oun
\item[\gfs{p'alin}] \palin
\item[\gfs{te\dots ka'i}] hem\dots hem
\item[\gfs{to'inun}] \toinun
\item[\gfs{<'wste}] \hoste, \hosteB
\end{description}

%\bibliographystyle{plain}
%\bibliography{../Public/references}
%\bibliography{../references}

\def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
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\end{thebibliography}


%\raggedright
%\printindex

\end{document} 
