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\begin{document}
\title{{\itshape{Toplama}}'n\i n\\
Yedinci Kitab\i'nda\\
\"Oklid'in \emph{Porizmalar}'\i\ \.I\c cin\\
38 Lemmadan\\
\.Ilk 19 Lemma\\
TASLAK%
%Pappus Alt\i gen Teoremi
}
\author{\.Iskenderiyeli Pappus\\
\c Ceviren: David Pierce}
\date{29 Eyl\"ul 2015}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\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 David Pierce \ccnc \ccsa\\
\mbox{}\\
\url{dpierce@msgsu.edu.tr}\\
}

\maketitle

\tableofcontents
%\listoffigures
\addchap{Giri\c s}

\addsec{Pascal Teoremi}

Bir alt\i genin k\"o\c seleri
bir koni kesitindeyse,
alt\i genin kenarlar\i\ 
\Sekilde{fig:el}ki gibi
\begin{figure}[ht]
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\caption{Elipste Alt\i gen Teoremi}\label{fig:el}
\end{figure}
uzat\i l\i nca,
kar\c s\i\ kenarlar\i n kesi\c sim noktalar\i\
bir do\u grudad\i r.
Alt\i gen d\i\c sb\"ukey olmayabilir;
kenarlar\i n kendileri zaten kesi\c sebilir.
\"Orne\u gin \Seklin{fig:hex} her \c s\i kk\i nda
\begin{figure}[h]
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\caption{Hiperbolde Alt\i gen Teoremi}\label{fig:hex}
\end{figure}
$A$, $B$, $C$, $D$, $E$, ve $F$ noktalar\i\
bir hiperbol\"un dallar\i ndad\i r,
ve (sonu\c c olarak) $G$, $H$, ve $K$ bir do\u grudad\i r.
Bu sonu\c c, \textbf{Alt\i gen Teoremi}
veya \textbf{Pascal Teoremi'dir:}
1640 y\i l\i nda, 16 ya\c s\i nda,
Blaise Pascal bu teoremi bildirdi.
(Pascal'in Frans\i zcas\i, \cite{Pascal-Coniques} kayna\u g\i ndad\i r;
\.Ingilizce \c cevirisi, \cite[s.\ 326--30]{MR0106139}
ve \cite[s.\ 163--8]{MR858706} kaynaklar\i ndad\i r.)

Pascal Teoremi'nde,
iki paralel do\u gru
\emph{sonsuzdaki noktada} kesi\c sen olarak say\i l\i r.
\"Orne\u gin \Sekilde{fig:hex},
$AB$ ve $DE$ birbirine paralel ise,
o zaman $HK$ de onlara paraleldir.

\addsec{Pappus Teoremi}

Pascal Teoremi'nde de
iki do\u gru,
bir koni kesiti olarak say\i l\i r.
\"Orne\u gin \Seklin{fig:hex} 
%\ref{fig:hex-a} 
(b) \c s\i kk\i nda,
$AEC$ ve $DBF$ do\u gru olabilir.
Teoremin bu durumu, \textbf{Pappus Teoremi'dir.}
\Seklin{fig:hex} 
%\ref{fig:hex-b} 
(a) \c s\i kk\i nda,
$AFD$ ve $BEC$ do\u gru ise, teorem a\c sik\^ard\i r.

%\"U\c c y\"uz 
M.S. 300
y\i l\i\ civar\i nda,
\emph{Toplama} (\gr{Sunagwg'h}, \emph{Collection}) adl\i\ eserinde
\.Iskenderiyeli Pappus,
ad\i n\i n verilece\u gi teoremini kan\i tlad\i.
\emph{Toplama}'n\i n sekiz kitab\i\ vard\i r.
Yedinci kitab\i,
\emph{Analiz Hazinesi}
(\gr{Analu'omenos t'opos}, \emph{Treasury of Analysis})
adl\i\ eserler i\c cin bir rehberdir.
Pappus'a g\"ore \cite[s.\ 598--601]{MR13:419b},
\emph{Hazine}'nin i\c cindekiler,
\sayfada{fig:treasury}ki gibidir.
Apollonius'un \emph{Koni Kesitleri} ve \"Oklid'in \emph{Veriler}'i hari\c c,
\emph{Hazine}'nin \c co\u gu \c simdi kaybolmu\c stur.
\"Ozel olarak \"Oklid'in \emph{Porizmalar}'\i\ kaybolmu\c stur.
Ama bu eserin okunmas\i na yard\i mc\i\ olmak i\c cin,
Pappus 38 lemmay\i\ verir.
Bunlar\i n ilk 19'nun \c cevirisi
a\c sa\u g\i dad\i r.
Pappus Alt\i gen Teoremi'nin baz\i\ durumlar\i, 
Lemma VIII, XII, ve XIII't\"ur.
Pappus, 
bug\"unk\"u kendisinin ad\i n\i\ alan
teoremin her durumunu kan\i tlamaz,
ama kan\i tlad\i\u g\i\ durumlar\i n kan\i tlar\i,
Lemma III, X, ve  XI'i kullan\i r.

Proklus'a g\"ore
(\cite[s.\ 236]{MR1200456} veya \cite[s.\ 478--81]{MR13:419a} kaynaklar\i nda),
\textbf{porizma} s\"ozc\"u\u g\"un\"un iki anlam\i\ vard\i r.
Porizman\i n birinci anlam\i,
kan\i t\i n\i n ba\c ska bir \"onermenin kan\i t\i ndan kolayca \c c\i kt\i\u g\i\
bir \"onermedir.
\"Orne\u gin
\"Oklid'in
\emph{\"O\u geler} Kitap VII'nin \"Onerme 2'si,
birbirine asal olmayan iki say\i n\i n en b\"uy\"uk ortak b\"olenini bulma problemidir.
Problemin \c c\"oz\"um\"unden,
say\i lar\i n her ortak b\"oleninin en b\"uy\"uk ortak b\"olenini b\"old\"u\u g\"u  
porizma \c c\i kar.

Porizma s\"ozc\"u\u g\"un\"un ikinci anlam\i na g\"ore
bir porizma, teorem ve problemin aras\i ndad\i r.
\"Orne\u gin
verilen a\c c\i y\i\ ikiye b\"olmek bir problemdir,
ama dairenin merkezini bulmak bir porizmad\i r,
\c c\"unk\"u bulunmadan \"once,
dairenin merkezinin var oldu\u gu bilinir.

D\"ort tane nokta verilirse,
alt\i\ tane do\u gru
noktalar\i n ikisinden ge\c cer.
Bu do\u grular\i n 
verilen bir do\u gruyu nas\i l kesti\u gi,
Pappus'un Lemma I, II, IV, V, VI, ve VII'sinin konusudur.
\"Orne\u gin
\Sekilde{fig:4}
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$A$, $B$, $C$, $D$, ve $E$ noktalar\i\
bir do\u gruda olsun.
\begin{compactenum}[(a)]
\item
Rasgele $F$ noktas\i\ se\c cilsin,
ve
$FA$, $FB$, ve $FC$ do\u grular\i\ birle\c stirilsin.
\item
$AF$ do\u grusunda rastgele $G$ noktas\i\ se\c cilsin,
ve
$GD$ ve $GE$ do\u grular\i\ birle\c stirilsin.
\item
S\i ras\i yla $FB$ ve $FC$'yi $H$ ve $K$'de kesince,
$HK$ do\u grusu birle\c stirilsin
ve $AB$'yi $L$'de kessin.
\end{compactenum}
O zaman Pappus'un g\"osterdi\u gine g\"ore
\begin{comment}


\Sekilde{fig:4-twice} gibi
\begin{figure}
\centering
\psset{unit=4mm}
\begin{pspicture}(0,0)(16,11)
\uput[d](0,0){$A$}
\uput[d](5.714,1.429){$B$}
\uput[d](10,2.5){$C$}
\uput[d](8,2){$D$}
\uput[d](12,3){$E$}
\psdots(0,0)(5.714,1.429)(10,2.5)(8,2)(12,3)
\uput[u](10,10){$F$}
\uput[ul](6,6){$G$}
\uput[l](7,4){$H$}
\uput[ur](10,4){$K$}
\uput[d](16,4){$L$}
\psline(0,0)(10,10)(5.714,1.429)
\psline(10,10)(10,2.5)
\psline(12,3)(6,6)(8,2)
\psline(7,4)(17,4)
\psline(0,0)(17,4.25)
\psdots[dotstyle=o](10,10)(6,6)(7,4)(10,4)(16,4)
\end{pspicture}
\caption{Ba\u g\i ms\i zl\i k}\label{fig:4-twice}
\end{figure}


\end{comment}
$L$ noktas\i,
$F$ ve $G$ noktalar\i n\i n se\c ciminden ba\u g\i ms\i zd\i r.


A\c sa\u g\i daki lemmalarda,
Pappus s\i k s\i k
orant\i lar\i\ kullan\i r.
Bunlar\i n kuram\i\
\c simdi \"ozetlenecektir.

\addsec{B\"uy\"ukl\"ukler ve katlar\i}

\"Oklid i\c cin
``e\c sit'' (\gr{>'isos}),
``ayn\i'' (\gr{a>ut'os})
de\u gildir.
\.Iki farkl\i\ s\i n\i rlanm\i\c s do\u gru
birbirine e\c sit olabilir.
E\c sitlik h\^al\^a
bir \emph{denklik ba\u g\i nt\i s\i d\i r.}
S\i n\i rlanm\i\c s bir do\u grunun \textbf{uzunlu\u gu,}
do\u grunun e\c sitlik s\i n\i f\i\ olarak
tan\i mlanabilir.
Benzer \c sekilde bir fig\"ur\"un \textbf{alan\i,}
fig\"ur\"un e\c sitlik s\i n\i f\i\ olarak
tan\i mlanabilir.

E\u ger \gr{AB} ve \gr{GD} do\u gru ise,
geni\c sli\u gi \gr{AB}'n\i n uzunlu\u gu olan
ve y\"uksekli\u gi \gr{GD}'n\i n uzunlu\u gu olan bir dikd\"ortgen in\c sa edilebilir,
ve bu dikd\"ortgeni
\begin{equation*}
\grm{AB}\cdot\grm{GD}
\end{equation*}
ifadesiyle yazaca\u g\i z;
Pappus
\begin{quote}
\centering
\gr{t`o <up`o AB GD}
\end{quote}
(``\gr{AB} ve \gr{GD} alt\i ndaki'')
ifadesini kullan\i yor.

\emph{Oranlar\i n} soyut kuram\i,
\emph{\"O\u geler}'in be\c sinci kitab\i ndad\i r.
\"Oklid'in tan\i mlad\i\u g\i\
her oran\i,
pozitif ger\c cel say\i\ olarak g\"orebiliriz
(ama tersi yanl\i\c s olabilir).

Bir \gr{AB} do\u grusunun,
bir \gr{GD} do\u grusuna oran\i\ vard\i r.
Bu oran\i
\begin{equation*}
\grm{AB}:\grm{GD}
\end{equation*}
olarak yazaca\u g\i z;
Pappus
\begin{quote}
\centering
\gr{l'ogos <`on >'eqei <h AB pr`os t`hn GD}
\end{quote}
(``\gr{AB}'n\i n \gr{GD}'ya oran\i,''
\emph{ratio that \gr{AB} has to \gr{GD}})
ifadesini kullan\i yor.

Ayn\i\ \c sekilde
bir $\grm{AB}\cdot\grm{GD}$ dikd\"ortgenin,
bir $\grm{EZ}\cdot\grm{HJ}$ dikd\"ortgenine oran\i\ vard\i r.
Bu oran\i
\begin{equation*}
\grm{AB}\cdot\grm{GD}:\grm{EZ}\cdot\grm{HJ}
\end{equation*}
olarak yazaca\u g\i z.

Her s\i n\i rlanm\i\c s do\u gru veya fig\"ur,
bir \textbf{b\"uy\"ukl\"ukt\"ur} (\gr{t`o meg'ejos}).
(Cisimler de b\"uy\"ukl\"ukt\"urler.)
Asl\i nda \"Oklid, iki b\"uy\"ukl\"u\u g\"un oran\i n\i\ (\gr{<o l'ogos}) tan\i mlamaz,
ama iki oran\i n ayn\i l\i\u g\i n\i\ (\gr{>an'alogon}) tan\i mlar.
Tan\i m a\c sa\u g\i daki gibidir.

Bir b\"uy\"ukl\"u\u g\"un \textbf{kat\i} (\gr{pollapl'asion}) al\i nabilir.
\"Orne\u gin \gr A bir b\"uy\"ukl\"uk ise,
katlar\i
\begin{align*}
	&\grm A,&
	&\grm A+\grm A,&
	&\grm A+\grm A+\grm A,&
	&\grm A+\grm A+\grm A+\grm A,
\end{align*}
ve sairedir.
Bunlar \gr A, $2\grm A$, $3\grm A$, ve saire olarak yaz\i labilir.
\gr A gibi b\"uy\"ukl\"ukt\"urler.
Asl\i nda \gr A'n\i n herhangi bir kat\i
\begin{equation*}
k\grm A
\end{equation*}
bi\c ciminde yaz\i labilir.
Buradaki $k$ katsay\i s\i,
do\u gal say\i d\i r.
Do\u gal say\i lar\i n olu\c sturdu\u gu k\"ume
\begin{equation*}
\N
\end{equation*}
olarak yaz\i labilir.
(Bizim i\c cin s\i f\i r, do\u gal say\i\ de\u gildir.)
%Biz,
$k$'n\i n do\u gal say\i\ oldu\u gunu g\"ostermek i\c cin
\begin{equation*}
k\in\N
\end{equation*}
ifadesini yazar\i z; 
ama \"Oklid bunun gibi bir ifade kullanmaz.
\"Oklid i\c cin, $k$ ifademiz isim de\u gil, s\i fat olurdu.
Asl\i nda \emph{\"O\u geler}'in yedinci kitab\i ndaki tan\i ma g\"ore
bir say\i\ (\gr{<o >arijm'os}), 
birimlerin (\gr{t`a mon'ada}) olu\c sturdu\u gu
bir \c cokluktur (\gr{t`o pl~hjos}).
Bu tan\i ma g\"ore bir $k\grm A$ kat\i n\i n kendisi bir say\i d\i r.
\"Oklid i\c cin
e\u ger \gr B \gr A'ya e\c sit olmayan bir b\"uy\"ukl\"uk ise,
o zaman 
$k\grm A$ ve $k\grm B$
b\"uy\"ukl\"uk olarak
birbirine e\c sit de\u gildir,
ama
\textbf{kat olarak birbirine e\c sittir,}
yani \textbf{e\c sit katlard\i r}
(\gr{>is'akis pollaplas'ia}).

\addsec{Oranlar ve orant\i l\i l\i k}

%\.Iki b\"uy\"ukl\"u\u g\"un e\c sit katlar\i, b\"uy\"ukl\"uk olarak e\c sit olmayabilir.
Herhangi \gr A ve \gr B b\"uy\"ukl\"ukleri kar\c s\i la\c st\i rabilir:
\gr A \gr B'dan b\"uy\"uk veya k\"u\c c\"uk olabilir, 
ve (b\"uy\"ukl\"uk olarak)
ikisi birbirine e\c sit olabilir.
S\i ras\i yla
\begin{align*}
	\grm A&>\grm B,&
	\grm A&<\grm B,&
	\grm A&=\grm B
\end{align*}
ifadelerini yazar\i z.
\.Iki b\"uy\"ukl\"u\u g\"un her biri do\u gru
veya her biri fig\"ur ise,
o zaman varsay\i ma g\"ore
b\"uy\"ukl\"uklerin her birinin bir kat\i,
\"oteki b\"uy\"ukl\"ukten b\"uy\"ukt\"ur.
Bu varsay\i ma
\textbf{Ar\c simet Aksiyomu} denir,
\c c\"unk\"u Ar\c simet onu yaz\i p kulland\i\
\cite[s.\ 36]{MR2093668};
ama Ar\c simet'ten \"once
 \"Oklid onu kulland\i.
\"Oklid, M.\"O. 300 civar\i nda \c cal\i\c s\i yordu;
Ar\c simet, M.\"O. 212 y\i l\i nda
Sirak\"uza'n\i n 
Romal\i lar taraf\i ndan al\i nmas\i nda \"old\"ur\"uld\"u.

E\u ger iki b\"uy\"ukl\"uk Ar\c simet Aksiyomunu sa\u glarsa,
\textbf{b\"uy\"ukl\"uklerin oran\i\ vard\i r.}
Bu \c sekilde \gr A ve \gr B'n\i n oran\i\ vard\i r
ancak ve ancak bir $k$ do\u gal say\i s\i\ i\c cin
$k\grm A>\grm B$ ve $\grm A<k\grm B$.


\c Simdi d\"ort tane b\"uy\"ukl\"uk al\i ns\i n.
Bunlardan
birincinin ve ikincinin oran\i\ olsun;\
\"u\c c\"unc\"un\"un ve d\"ord\"unc\"un\"un de oran\i\ olsun.
B\"uy\"ukl\"uklerden birincinin ve \"u\c c\"unc\"un\"un herhangi e\c sit kat\i\
ve ikincinin ve d\"ord\"unc\"un\"un herhangi e\c sit kat\i\ al\i n\i nca,
e\u ger birincinin ve \"u\c c\"unc\"un\"un katlar\i\
s\i ras\i yla
ayn\i\ zamanda 
\begin{inparaenum}[(1)]
\item
ya ikincinin ve d\"ord\"unc\"un\"un katlar\i ndan b\"uy\"uk,
\item
ya ikincinin ve d\"ord\"unc\"un\"un katlar\i na e\c sit,
\item
ya da ikincinin ve d\"ord\"unc\"un\"un katlar\i ndan k\"u\c c\"uk ise,
\end{inparaenum}
o zaman \"Oklid'in tan\i m\i na g\"ore
birincinin ikincine \textbf{oran\i,}
\"u\c c\"unc\"un\"un d\"ord\"unc\"une \textbf{oran\i yla ayn\i d\i r,}
ve d\"ort b\"uy\"ukl\"uk \textbf{orant\i l\i d\i r.}

Asl\i nda b\"uy\"ukl\"ukler \gr A, \gr B, \gr G, ve \gr D olsun.
E\u ger herhangi $k$ ve $m$ do\u gal say\i lar\i\ i\c cin
\begin{gather*}
	k\grm A>m\grm B\iff k\grm G>m\grm D,\\
	k\grm A=m\grm B\iff k\grm G=m\grm D,\\
	k\grm A<m\grm B\iff k\grm G<m\grm D
\end{gather*}
denklikleri do\u gru ise, 
o zaman
\gr A, \gr B, \gr G, ve \gr D orant\i l\i d\i r,
ve $\grm A:\grm B$ ve $\grm G:\grm D$ oranlar\i\
birbiriyle ayn\i d\i r.
Bu durumda
\begin{equation*}
\grm A:\grm B::\grm G:\grm D
\end{equation*}
ifadesini yazar\i z.
%bu bir \textbf{orant\i d\i r.}
\"Oklid ve Pappus sadece s\"ozler kullan\i rlar.
\"Orne\u gin \gr{AB}'n\i n \gr{GD}'ya oran\i,
$\grm{EZ}\cdot\grm{HJ}$'n\i n
$\grm{KL}\cdot\grm{MN}$'ye oran\i yla ayn\i ysa
\begin{equation*}
\grm{AB}:\grm{GD}::\grm{EZ}\cdot\grm{HJ}:\grm{KL}\cdot\grm{MN}
\end{equation*}
ifadesini yazaca\u g\i z,
ama Pappus
\begin{quote}
\centering
\gr{<ws <h AB pr`os <h GD,}\\
\gr{o<'utws t`o <up`o EZ HJ pr`os t`o <up`o KL MN}
\end{quote}
yaz\i yor.
(Pappus'un metnini bulundu\u gumuz el yazmas\i nda,
k\i saltmalar kullan\i l\i r,
ama matematik kavramlar\i\ de\u gil,
sadece s\"ozc\"ukler i\c cin
\cite[p.\ 28]{Jones}.)

Herhangi \gr A, \gr B, \gr G, ve \gr D b\"uy\"ukl\"ukleri i\c cin
\begin{equation*}
\grm A:\grm B::\grm G:\grm D
\end{equation*}
olsun.
Oranlar\i n ayn\i l\i\u g\i\
bir denklik ba\u g\i nt\i s\i\ olunca
\begin{equation*}
\grm G:\grm D::\grm A:\grm B
\end{equation*}
da do\u grudur.
\emph{\"O\u geler} Kitap V'e g\"ore
\begin{itemize}
\item
\textbf{\anapalin} (\gr{>an'apalin})
\begin{math}
\grm B:\grm A::\grm D:\grm G,
\end{math}
\item
\textbf{\synthenti} (\gr{sunj'enti})
\begin{math}
\grm A+\grm B:\grm B::\grm G+\grm D:\grm D,
\end{math}
\item
\textbf{\anastrepsanti} (\gr{>anastr'eyant'i})
\begin{math}
\grm A-\grm B:\grm B::\grm G-\grm D:\grm D
\end{math}
(burada $\grm A>\grm B$ olmal\i),
\item
\textbf{\enallax} (\gr{>enall'ax})
\begin{math}
\grm A:\grm G::\grm B:\grm D
\end{math}
(burada \gr A'n\i n \gr G'ya ve \gr B'n\i n \gr D'ya oran\i\ var olmal\i).
\end{itemize}
Ayr\i ca, e\u ger
$\grm A:\grm B::\grm D:\grm E$
ve $\grm B:\grm G::\grm E:\grm Z$
%\begin{align*}
%	\grm A:\grm B&::\grm D:\grm E,&
%	\grm B:\grm G&::\grm E:\grm Z
%\end{align*} 
ise,
o zaman
\textbf{\diisou} (\gr{di' >'isou})
\begin{equation*}
\grm A:\grm E::\grm G:\grm Z.
\end{equation*}
Daha genelde, e\u ger
$\grm A_1$, \dots, $\grm A_n$ ve
$\grm A_1$, \dots, $\grm B_n$ 
b\"uy\"ukl\"ukleri verilirse,
ve her durumda
\begin{equation*}
\grm A_k:\grm A_{k+1}::\grm B_k:\grm B_{k+1}
\end{equation*}
ise,
o zaman \diisou
\begin{equation*}
\grm A_1:\grm A_n::\grm B_1:\grm B_n;
\end{equation*}
herhalde \"Oklid ``\diisou'' diyor
\c c\"unk\"u \c cokluklar olarak
$A_k$'lerin ve $B_k$'lerin say\i lar\i\
%$\{\grm A_1,\dots,\grm A_n\}$ ve
%$\{\grm B_1,\dots,\grm B_n\}$ k\"umeleri
birbirine e\c sittir.

\c Simdi \gr A, \gr B, ve \gr G do\u gru olsun.
O zaman
\emph{\"O\u geler} Kitap VI'n\i n \"Onerme 1'ine g\"ore
\begin{equation*}
\grm A:\grm B::\grm A\cdot\grm G:\grm B\cdot\grm G.
\end{equation*}
Ayr\i ca,
tan\i ma g\"ore
$\grm A:\grm B$ ve
$\grm B:\grm G$ oranlar\i n\i n \textbf{bile\c simi} 
(\gr{sunhmm'enos},
\gr{sun'aptw} f\/iilinden),
$\grm A:\grm G$ oran\i d\i r,
ama bu tan\i m \emph{\"O\u geler}'de de\u gildir.
E\u ger $\grm B:\grm G::\grm D:\grm E$ ise,
o zaman
$\grm A:\grm G$ oran\i,
$\grm A:\grm B$ ve
$\grm D:\grm E$ oranlar\i n\i n bile\c simiyle ayn\i d\i r.
Bu orant\i y\i\
\begin{equation*}
\grm A:\grm G::\grm A:\grm B\And \grm D:\grm E
\end{equation*}
bi\c ciminde yazaca\u g\i z.
\"Orne\u gin a\c sa\u g\i daki Lemma I'de Pappus
\begin{quote}
\centering
\gr{<o t~hs AD pr`os t`hn DZ sun~htai\\
>'ek te to~u t~hs AB pr`os t`hn BE\\ka`i to~u t~hs EJ pr`os JH}
\end{quote}
orant\i s\i n\i\ yaz\i yor; bunu
\begin{equation*}
\grm{AD}:\grm{DZ}::\grm{AB}:\grm{BE}\And\grm{EJ}:\grm{JH}
\end{equation*}
olarak yazaca\u g\i z.
Lemma III'te Pappus
\begin{quote}
\centering
\gr{to~u <up`o JE HZ pr`os t`o <up`o JH ZE sun~htai l'ogos\\
>'ek te to~u <`on >'eqei <h JE pr`os t`hn EZ\\
ka`i to~u <`on >'eqei <h ZH pr`os t`hn HJ}
\end{quote}
orant\i s\i n\i\ yaz\i yor; bunu
\begin{equation*}
\grm{JE}\cdot\grm{HZ}:\grm{JH}\cdot\grm{ZE}::\grm{JE}:\grm{EZ}\And\grm{ZH}:\grm{HJ}
\end{equation*}
olarak yazaca\u g\i z.

Son olarak, \Sekilde{fig:vi.2},
\begin{figure}[ht]
\centering
\psset{unit=10mm}
\begin{pspicture}(0,-0.5)(4,4.5)
\pspolygon(0,0)(4,0)(2,4)
\psline(1,2)(3,2)
\uput[u](2,4){\gr A}
\uput[d](0,0){\gr B}
\uput[d](4,0){\gr G}
\uput[ul](1,2){\gr D}
\uput[ur](3,2){\gr E}
\end{pspicture}
\caption{\emph{\"O\u geler} \textsc{vi}.2}\label{fig:vi.2}
\end{figure}
e\u ger \gr{DE} \gr{BG}'ya paralel ise,
o zaman
\begin{equation*}
\grm{BD}:\grm{DA}::\grm{GE}:\grm{EA};
\end{equation*}
ve tersi de do\u grudur.
Bu sonu\c c,
\emph{\"O\u geler} Kitap VI'n\i n \"Onerme 2'sidir.


\addsec{Pappus \c cevirisi hakk\i nda}

A\c sa\u g\i daki \c ceviri i\c cin
Hultsch'un \cite{Pappus} edisyonunu kulland\i m.
Sadece bittikten sonra Jones'un \cite{Jones} edisyonunu
bulup onunla yapt\i\u g\i m\i\ d\"uzelttim.

Hultsch, \emph{Toplama} Kitap VII'nin t\"um \"onermelerine
Arap rakamlar\i\ koyar,
ve oradaki 
\"Oklid'in \emph{Porizmalar}'\i\ hakk\i ndaki lemmalara
Romen rakamlar\i\ koyar.
Yukar\i da bahsetti\u gimiz gibi
Pappus, oran ve orant\i lar i\c cin \"ozel i\c saretler kullanmaz.
Ayr\i ca \c sekillerinde
t\"um \c cizgilerin kal\i nl\i\u g\i\ ayn\i d\i r.
Lemmalar\i n kan\i tlar\i n\i\
``\emph{Kan\i t}'' ve $\qedsymbol$ aras\i nda yaz\i yorum;
Pappus bunun gibi ifadeler kullanmaz.



\addpart[Pappus'tan 19 Lemma]%
{Pappus'un \emph{Toplama}'s\i n\i n\\
Yedinci Kitab\i'nda\\
\"Oklid'in \emph{Porizmalar}'\i\ \.I\c cin\\
38 Lemmas\i ndan\\
\.Ilk 19 Lemma}

%\nextlem

\addchap{Lemma I (\"Onerme 127)}

\begin{theorem}
Diyagram \gr{ABGDEZH} olsun
 (\gr{>'Estw katagraf`h <h ABGDEZH}),
ve
\begin{equation*}
\grm{AZ}:\grm{ZH}::\grm{AD}:\grm{DG}
\end{equation*}
olsun,
ve \gr{JK} birle\c stirilmi\c s olsun.
\Hoti
\begin{equation*}
\text{\gr{JK}, \gr{AG}'ya paraleldir.}
\end{equation*}
\end{theorem}
\begin{figure}
\centering
\psset{unit=5mm}
\begin{pspicture}(0,-1)(12,9)
\pspolygon[linewidth=2pt](4,2)(8,4)(10,8)(4,8)
\psline[linewidth=2pt](0,0)(12,0)
\psline(0,0)(4,2)
\psline(4,0)(4,2)
\psline(2,0)(10,8)
\psline(6,0)(8,4)
\psline(4,8)(12,0)
\psset{linestyle=dotted}
\psline(6,0)(12,6)
\psline(12,0)(12,6)(8,4)
\uput[d](0,0){\gr A}
\uput[ul](4,2){\gr B}
\uput[d](4,0){\gr G}
\uput[d](2,0){\gr D}
\uput[u](8,4){\gr E}
\uput[d](6,0){\gr Z}
\uput[d](12,0){\gr H}
\uput[ul](4,8){\gr J}
\uput[ur](10,8){\gr K}
\uput[ur](12,6){\gr L}
\end{pspicture}
\end{figure}

\begin{proof}
\gr Z'dan \gr{BD}'ya paralel olan \gr{ZL} \echthw\ olsun.
\Oun
\begin{equation*}
\grm{AZ}:\grm{ZH}::\grm{AD}:\grm{DG}
\end{equation*}
oldu\u gundan
\anapalin\ ve \synthenti\ ve \enallax\
(ve parallellerden)
\begin{equation*}
\underbrace{\grm{DA}:\grm{AZ}}_{\displaystyle\grm{BA}:\grm{AL}}
::\grm{GA}:\grm{AH}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{LH}, \gr{BG}'ya paraleldir.}
\end{equation*}
\Ara\ (paralellerden)
\begin{equation*}
	\grm{EB}:\grm{BL}::
\left\{
\begin{gathered}
        \grm{EK}:\grm{KZ}\\
	\grm{EJ}:\grm{JH}.
\end{gathered}\right.
\end{equation*}
\Ara
\begin{equation*}
\grm{EK}:\grm{KZ}::\grm{EJ}:\grm{JH}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{JK}, \gr{AG}'ya paraleldir.}\qedhere
\end{equation*}
\end{proof}

\begin{proof}[Bile\c sik oranlar\i\ kullanan kan\i t.]
\addcontentsline{toc}{section}{\.Ikinci kan\i t\i}
\begin{equation*}
\grm{AZ}:\grm{ZH}::\grm{AD}:\grm{DG}
\end{equation*}
oldu\u gundan,
\anapalin
\begin{equation*}
\grm{HZ}:\grm{ZA}::\grm{GD}:\grm{DA}.
\end{equation*}
\Synthenti\ ve \enallax\ ve \anastrepsanti
\begin{equation*}
\grm{AD}:\grm{DZ}::\grm{AG}:\grm{GH}.
\end{equation*}
\Alla\
\begin{equation*}
\grm{AD}:\grm{DZ}::
\grm{AB}:\grm{BE}\And\grm{EJ}:\grm{JH},
\end{equation*}
\ara
\begin{equation*}
\grm{AB}:\grm{BE}\And\grm{EK}:\grm{KZ}
::
\grm{AB}:\grm{BE}\And\grm{EJ}:\grm{JH}.
\end{equation*}
Ortak $\grm{AB}:\grm{BE}$ oran\i\ kovulmu\c s olsun.
\Ara\ kalan
\begin{equation*}
\grm{EK}:\grm{KZ}
::
\grm{EJ}:\grm{JH}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{JK}, \gr{AG}'ya paraleldir.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma II (\"Onerme 128)}

\begin{theorem}
Diyagram \gr{ABGDEZHJ} [olsun],
ve \gr{AZ} \gr{DB}'ya paralel olsun, ve
\begin{equation*}
\grm{AE}:\grm{EZ}::\grm{GH}:\grm{HZ}
\end{equation*}
[olsun].
\Hoti
\begin{equation*}
\text{\gr J, \gr K, ve \gr Z'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=5mm}
\begin{pspicture}(0,-0.75)(12,8.75)
\pspolygon[linewidth=2pt](4,2)(8,4)(10,8)(4,8)
\psline[linewidth=2pt](0,0)(12,0)
\psline(0,0)(4,2)
\psline(4,0)(4,2)
\psline(2,0)(10,8)
\psline(6,0)(8,4)
\psline(4,8)(12,0)
\psset{linestyle=dotted}
\psline(2,0)(2.667,1.333)
\uput[d](0,0){\gr Z}
\uput[ul](4,2){\gr K}
\uput[d](4,0){\gr G}
\uput[d](2,0){\gr H}
\uput[r](8,4){\gr J}
\uput[d](6,0){\gr E}
\uput[d](12,0){\gr A}
\uput[ul](4,8){\gr B}
\uput[ur](10,8){\gr D}
\uput[ul](2.667,1.333){\gr L}
\end{pspicture}
\end{figure}

\begin{proof}
\gr H'dan \gr{DE} boyunca \gr{HL} \echthw\ olsun,
ve birle\c stirilmi\c s olan \gr{JK}
\gr L'ya uzat\i lm\i\c s olsun.
\begin{equation*}
\grm{AE}:\grm{EZ}::\grm{GH}:\grm{HZ}
\end{equation*}
oldu\u gundan,
\enallax
\begin{equation*}
\grm{AE}:\grm{GH}::\grm{EZ}:\grm{ZH}.
\end{equation*}
Ayr\i ca
\begin{equation*}
\grm{AE}:\grm{GH}::\grm{EJ}:\grm{HL}
\end{equation*}
(\c c\"unk\"u iki do\u gru iki do\u gruya paralel,
ve \enallax).
\Ara
\begin{equation*}
\grm{EZ}:\grm{ZH}::\grm{EJ}:\grm{HL}.
\end{equation*}
Ayr\i ca \gr{EJ} \gr{HL}'ya paraleldir.
\Ara
\begin{equation*}
\text{\gr J, \gr L, \gr Z'dan [ge\c cen \c cizgi] do\u grudur,}
\end{equation*}
yani \gr J, \gr K, \gr Z'dan.
\end{proof}

%\nextlem

\addchap{Lemma III (\"Onerme 129)}

\begin{theorem}
\"U\c c do\u gru \gr{AB}, \gr{GA}, ve \gr{DA} \"uzerine
iki do\u gru \gr{JE} ve \gr{JD} \diechthw\ olsun.
\Hoti
\begin{equation*}
\grm{JE}\cdot\grm{HZ}:\grm{JH}\cdot\grm{ZE}::\grm{JB}\cdot\grm{DG}:\grm{JD}\cdot\grm{BG}.
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=1.8cm}
\begin{pspicture}(-2,-1)(4,2)
%\psgrid
  \psline[linewidth=2pt](1,0)(2,1)(4,0)
\psline[linewidth=2pt](2,0)(2,1)
\psline(2.667,0.667)(0,0)(4,0)
\psset{linestyle=dotted}
\psline(2,1)(0,2)(0,-1)
\psline(1,0)(0,-1)(-2,0)(0,0)
%\pspolygon(0,0)(-2,0)(0,-1)
\psdot(0,0)
\uput[ur](2,1){\gr A}
\uput[ul](1.333,0.333){\gr B}
\uput[dr](2,0.5){\gr G}
\uput[ur](2.667,0.667){\gr D}
\uput[-67.5](1,0){\gr E}
\uput[d](2,0){\gr Z}
\uput[r](4,0){\gr H}
\uput[ul](0,0){\gr J}
\uput[l](0,2){\gr K}
\uput[r](0,-1){\gr L}
\uput[u](-2,0){\gr M}
\end{pspicture}
%\caption{Lemma III}
\end{figure}

\begin{proof}\mbox{}
\begin{itemize}
\item
\gr J'dan ge\c cen ve \gr{ZGA}'ya paralel olan
\gr{KL} \echthw\ olsun,
ve \gr{DA} ve \gr{AB} bununla kesi\c smi\c s olsun
\gr K ve \gr L noktalar\i nda;
\item
\gr L'dan da ge\c cen ve \gr{DA}'ya paralel olan
\gr{LM} de [\echthw\ olsun],
ve \gr{EJ} ile kesi\c smi\c s olsun \gr M'de.
\end{itemize}
\Oun,
\begin{align*}
\grm{EZ}:\grm{ZA}&::\grm{EJ}:\grm{JL},\\
\grm{AZ}:\grm{ZH}&::\grm{JL}:\grm{JM}
\end{align*}
(\c c\"unk\"u ikisi, $\grm{JK}:\grm{JH}$ ile ayn\i,
parallerden),
[ve bunlar] oldu\u gundan,
\ara\
\diisou
\begin{equation*}
\grm{EZ}:\grm{ZH}::\grm{EJ}:\grm{JM}.
\end{equation*}
\Ara
\begin{equation*}
\grm{JE}\cdot\grm{HZ}=\grm{EZ}\cdot\grm{JM}.
\end{equation*}
\Allode{EZ}{JH}.
\Ara
\begin{align*}
	\grm{EJ}\cdot\grm{HZ}:\grm{EZ}\cdot\grm{HJ}
	&::\grm{EZ}\cdot\grm{JM}:\grm{EZ}\cdot\grm{HJ}\\
	&::\grm{JM}:\grm{JH}\\
	&::\grm{LJ}:\grm{JK}.
\end{align*}
\Katatauta
\begin{equation*}
\grm{KJ}:\grm{JL}::\grm{JD}\cdot\grm{BG}\cdot\grm{JB}\cdot\grm{GD}.
\end{equation*}
\Ara\ \anapalin
\begin{equation*}
\grm{LJ}:\grm{JK}::\grm{JB}\cdot\grm{GD}:\grm{JD}\cdot\grm{BG}.
\end{equation*}
Ve
\begin{equation*}
\grm{LJ}\cdot\grm{JK}::\grm{EJ}\cdot\grm{HZ}:\grm{EZ}\cdot\grm{HJ}
\end{equation*}
g\"osterilmi\c s oldu.
Ve \ara
\begin{equation*}
\grm{EJ}\cdot\grm{HZ}:\grm{EZ}\cdot\grm{HJ}::\grm{JB}\cdot\grm{GD}:\grm{JD}\cdot\grm{BG}.\qedhere
\end{equation*}
\end{proof}

\clearpage

%\newpage

\begin{proof}[Bile\c sik oranlar\i\ kullanan kan\i t.]\FloatBarrier
\addcontentsline{toc}{section}{\.Ikinci kan\i t\i}
\begin{figure}
\centering
\psset{xunit=1.8cm,yunit=1.8cm}
\begin{pspicture}(-2,-1)(4,2)
%\psgrid
  \psline[linewidth=2pt](1,0)(2,1)(4,0)
\psline[linewidth=2pt](2,0)(2,1)
\psline(2.667,0.667)(0,0)(4,0)
\psset{linestyle=dotted}
\psline(2,1)(0,2)(0,-1)(1,0)
\pspolygon(0,0)(-2,0)(0,-1)
\psdot(0,0)
\uput[ur](2,1){\gr A}
\uput[ul](1.333,0.333){\gr B}
\uput[dr](2,0.5){\gr G}
\uput[ur](2.667,0.667){\gr D}
\uput[-67.5](1,0){\gr E}
\uput[d](2,0){\gr Z}
\uput[r](4,0){\gr H}
\uput[ul](0,0){\gr J}
\uput[l](0,2){\gr K}
\uput[r](0,-1){\gr L}
\uput[u](-2,0){\gr M}
\end{pspicture}
%\caption{Lemma III}
\end{figure}
\begin{gather*}
\grm{JE}\cdot\grm{HZ}:\grm{JH}\cdot\grm{ZE}::
\grm{JE}:\grm{EZ}\And\grm{ZH}:\grm{HJ},\\
	\grm{JE}:\grm{EZ}::\grm{JA}:\grm{ZA},\\
	\grm{ZH}:\grm{HJ}::\grm{ZA}:\grm{JK}
\end{gather*}
oldu\u gundan,
\ara\
\begin{equation*}
\grm{JE}\cdot\grm{HZ}:\grm{JH}\cdot\grm{EZ}::
\grm{JL}:\grm{ZA}\And\grm{ZA}:\grm{JK}.
\end{equation*}
Ayr\i ca
\begin{equation*}
\grm{JL}:\grm{ZA}\And\grm{ZA}:\grm{JK}::\grm{JL}:\grm{JK}.
\end{equation*}
\Ara
\begin{equation*}
\grm{JE}\cdot\grm{HZ}:\grm{JH}\cdot\grm{ZE}::\grm{JL}:\grm{JK}.
\end{equation*}
\Diatauta
\begin{equation*}
\grm{JD}\cdot\grm{BG}:\grm{JB}\cdot\grm{GD}::\grm{JK}:\grm{JL}.
\end{equation*}
Ve \anapalin
\begin{equation*}
\grm{JB}\cdot\grm{GD}:\grm{JD}\cdot\grm{BG}::\grm{JL}:\grm{JK}.
\end{equation*}
\Alla
\begin{equation*}
\grm{JE}\cdot\grm{ZH}:\grm{JH}\cdot\grm{ZE}::\grm{JL}:\grm{JK}
\end{equation*}
oldu.  
Ve \ara
\begin{equation*}
\grm{JE}\cdot\grm{ZH}:\grm{JH}\cdot\grm{ZE}::\grm{JB}\cdot\grm{GD}:\grm{JD}\cdot\grm{BG}.\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma IV (\"Onerme 130)}

\begin{theorem}
Diyagram \gr{ABGDEZHJKL} [olsun],
ve
\begin{equation*}
\grm{AZ}\cdot\grm{BG}:\grm{AB}\cdot\grm{GZ}
::\grm{AZ}\cdot\grm{DE}:\grm{AD}\cdot\grm{EZ}
\end{equation*}
olsun.
\Hoti
\begin{equation*}
\text{\gr J, \gr H, ve \gr Z
noktalar\i ndan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=6mm}
\begin{pspicture}(0,-1)(16,11)
\pspolygon[linewidth=2pt](10,4)(7,4)(6,6)(10,10)
\pspolygon[linewidth=2pt](0,0)(16,4)
\psline(5.714,1.429)(10,10)
\psline(6,6)(0,0)
\psline[linestyle=dotted](16,4)(10,4)
\psline(8,2)(6,6)(12,3)
\psline[linestyle=dotted](12,3)(11.5,4)
\psline(10,10)(10,2.5)
\psline[linestyle=dotted](6,6)(10,7)
\uput[d](0,0){\gr A}
\uput[d](5.714,1.429){\gr B}
\uput[d](8,2){\gr G}
\uput[d](10,2.5){\gr D}
\uput[d](12,3){\gr E}
\uput[d](16,4){\gr Z}
\uput[ur](10,4){\gr H}
\uput[l](7,4){\gr J}
\uput[ul](6,6){\gr K}
\uput[u](10,10){\gr L}
\uput[r](10,7){\gr M}
\uput[ul](8.286,6.571){\gr N}
\uput[u](11.5,4){\gr{\myXi}}
\end{pspicture}
\end{figure}

\begin{proof}
\begin{math}
\grm{AZ}\cdot\grm{BG}:\grm{AB}\cdot\grm{GZ}
::\grm{AZ}\cdot\grm{DE}:\grm{AD}\cdot\grm{EZ}
\end{math}
oldu\u gundan
\enallax
\begin{equation*}
\underbrace{\grm{AZ}\cdot\grm{BG}:\grm{AZ}\cdot\grm{DE}}
_{\displaystyle\grm{BG}:\grm{DE}}
::\grm{AB}\cdot\grm{GZ}:\grm{AD}\cdot\grm{EZ}.
\end{equation*}
\Alla\
(e\u ger \gr K'dan \gr{AZ}'ya paralel olan \gr{KM}
\echthw\ ise)
\begin{gather*}
	\grm{BG}:\grm{DE}::\grm{BG}:\grm{KN}\And\grm{KN}:\grm{KM}\And\grm{KM}:\grm{DE},\\
\grm{AB}\cdot\grm{GZ}:\grm{AD}\cdot\grm{EZ}
::\grm{BA}:\grm{AD}\And\grm{GZ}:\grm{ZE}.
\end{gather*}
$\grm{NK}:\grm{KM}$ ile ayn\i\ olan
ortak $\grm{BA}:\grm{AD}$ kovulmu\c s olsun,
\Ara\
\begin{equation*}
\text{kalan $\grm{GZ}:\grm{ZE}$}::
\underbrace{\grm{BG}:\grm{KN}}
_{\displaystyle\grm{JG}:\grm{KJ}}
\And
\underbrace{\grm{KM}:\grm{DE}}
_{\displaystyle\grm{KH}:\grm{HE}.}
\end{equation*}
\Ara
\begin{equation*}
\text{\gr J, \gr H, ve \gr Z'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\Gar\ e\u ger 
\gr E'dan \gr{JG}'ya paralel olan
\gr{E\myXi}'yi ilerlersem,
ve birle\c stirilmi\c s olan \gr{JH} \gr{\myXi}'ye uzat\i lm\i\c s olursa,
\begin{itemize}
\item
$\grm{KH}:\grm{HE}::\grm{KJ}:\grm{E\myXi}$,
\item
$\grm{GJ}:\grm{JK}\And\grm{JK}:\grm{E\myXi}$ bile\c simi
$\grm{JG}:\grm{E\myXi}$ oran\i yla de\u gi\c stirilir,
ve
\begin{equation*}
\grm{GZ}:\grm{ZE}::\grm{GJ}:\grm{E\myXi}.
\end{equation*}
\end{itemize}
\grm{GJ} \gr{E\myXi}'ye paralel olunca,
\ara\ 
\begin{equation*}
\text{\gr J, \gr{\myXi}, ve \gr Z'dan [ge\c cen \c cizgi] do\u grudur}
\end{equation*}
(\gar\ bu apa\c c\i kt\i r),
\hoste\ ayr\i ca
\begin{equation*}
\text{\gr J, \gr H, ve \gr Z'dan [ge\c cen \c cizgi] do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma V (\"Onerme 131)}

\begin{theorem}
E\u ger diyagram \gr{ABGDEZHJ} ise
[ve \"ozel olarak \gr A, \gr H, ve \gr J'dan ge\c cen \c cizgi do\u gru ise],
\begin{equation*}
\grm{AD}:\grm{DG}::\grm{AB}:\grm{BG}
\end{equation*}
meydana gelir.
\Oun\
\begin{math}
\grm{AD}:\grm{DG}::\grm{AB}:\grm{BG}
\end{math}
olsun.
\Hoti
\begin{equation*}
\text{\gr A, \gr H, ve \gr J'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=1.5mm}
\begin{pspicture}(0,-2.5)(60,32.5)
\pspolygon[linewidth=2pt](24,12)(30,15)(30,30)(20,20)
\psline[linewidth=2pt](0,0)(60,0)
\psline(20,0)(30,30)
\psline(20,20)(60,0)
\psline(30,15)(30,0)(24,12)
\psline(0,0)(20,20)
\psline[linestyle=dashed](24,12)(0,0)
\uput[d](0,0){\gr A}
\uput[d](20,0){\gr B}
\uput[d](30,0){\gr G}
\uput[d](60,0){\gr D}
\uput[u](30,30){\gr E}
\uput[ul](20,20){\gr Z}
\uput[-30](24,12){\gr H}
\uput[ur](30,15){\gr J}
\psline[linestyle=dotted](12,12)(36,12)
\uput[ul](12,12){\gr K}
\uput[ur](36,12){\gr L}
\uput[dr](30,12){\gr M}
\end{pspicture}
\end{figure}

\begin{proof}
\gr H'dan \gr{AD}'ya paralel olan \gr{KL} \echthw\ olsun.
\Oun
\begin{equation*}
\grm{AD}:\grm{DG}::\grm{AB}:\grm{BG}
\end{equation*}
oldu\u gundan,
\alla
\begin{gather*}
	\grm{AD}:\grm{DG}::\grm{KL}:\grm{LH},\\
	\grm{AB}:\grm{BG}::\grm{KH}:\grm{HM}
\end{gather*}
oldu\u gundan,
\ara\ ayr\i ca
\begin{equation*}
\grm{KL}:\grm{LH}::\grm{KH}:\grm{HM},
\end{equation*}
ve
\begin{align*}
\text{kalan \gr{HL}}:\text{kalan \gr{LM}}
&::\grm{KL}:\grm{LH}\\
	&::\grm{AD}:\grm{DG}.
\end{align*}
\Enallax
\begin{align*}
	\grm{AD}:\grm{HL}
	&::\grm{GD}:\grm{LM}\\
	&::\grm{DJ}:\grm{JL},
\end{align*}
ve \gr{HL} \gr{AD}'ya paraleldir.
\Ara
\begin{equation*}
\text{\gr A, \gr H, ve \gr J noktalar\i ndan
[ge\c cen \c cizgi] do\u grudur;}
\end{equation*}
\gar\ bu apa\c c\i kt\i r.
\end{proof}

%\nextlem

\addchap{Lemma VI (\"Onerme 132)}

\begin{theorem}
\Palin\ e\u ger diyagram [\gr{ABGDEZH}] ise,
ve \gr{DZ} \gr{BG}'ya paralel ise,
\begin{equation*}
\grm{AB}=\grm{BG}
\end{equation*}
meydana gelir.
\Oun\ e\c sit olsun;
\hoti
\begin{equation*}
\text{[\gr{DZ} \gr{BG}'ya] paraleldir.}
\end{equation*} 
\end{theorem}

\begin{figure}
\centering
%\psset{unit=5mm}
\begin{pspicture}(0,-2.5)(6,6.5)
\pspolygon[linewidth=2pt](4,2)(6,3)(6,6)(3,3)
\psline(6,6)(3,0)
\psline[linestyle=dotted](3,0)(2,-2)
\psline[linestyle=dotted](0,0)(2,-2)(6,0)
\pspolygon(3,3)(6,3)(6,0)(4,2)(0,0)
\psline[linewidth=2pt](0,0)(6,0)
\uput[l](0,0){\gr A}
\uput[dr](3,0){\gr B}
\uput[r](6,0){\gr G}
\uput[ul](3,3){\gr D}
\uput[u](6,6){\gr E}
\uput[r](6,3){\gr Z}
\uput[-75](4,2){\gr H}
\uput[d](2,-2){\gr J}
\end{pspicture}
\end{figure}

\begin{proof}
Olur da.
\Gar\ e\u ger \gr{EB}'da
\gr{HB}'ya e\c sit olan \gr{BJ}'y\i\ koyarsam,
ve \gr{AJ} ve \gr{JG}'y\i\ birle\c stirirsem,
\begin{equation*}
\text{paralelkenar \gr{AJGH} meydana gelir,}
\end{equation*}
ve bundan
\begin{equation*}
\grm{AD}:\grm{DE}::\grm{GZ}:\grm{ZE}
\end{equation*}
(\gar\ s\"oylenmi\c s [iki oran\i n] her biri $\grm{JH}:\grm{HE}$ oran\i yla ayn\i d\i r).
\Hoste\ 
\begin{equation*}
\text{\gr{DZ} \gr{AG}'ya paraleldir.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma VII (\"Onerme 133)}

\begin{theorem}
Diyagram olsun, 
ve \gr{DB} ve \gr{BG}'n\i n orta orant\i l\i s\i\ \gr{BA} olsun
(\gr{t~wn DB BG m'esh >an'alogon >'estw <h BA}).%%%%%
\footnote{Yani $\grm{DB}:\grm{BA}::\grm{BA}:\grm{BG}$ olsun.
\emph{\"O\u geler} \"Onerme VI.13'\"une bak\i n.
Yunan \gr{<h m'esh >an'alogon} teriminin \.Ingilizcesi
\emph{mean proportional}'d\i r,
\"orne\u gin lisede kulland\i\u g\i m 
geometri ders kitab\i nda \cite[s.\ 239]{Weeks-Adkins}.
T\"urk\c ce'de Demirta\c s
``orta orant\i l\i'' kullan\i r \cite[s.\ 214]{Demirtas},
ama Atat\"urk ``ortakoran'' kulland\i\ \cite[s.\ 35]{Ataturk}.}
%%%%%%%%%%%%%%%%%%%%%%%
\Hoti
\begin{equation*}
\text{\gr{ZH} \gr{AG}'ya paraleldir.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=1.5mm}
\begin{pspicture}(0,-12.5)(60,42.5)
\pspolygon[linewidth=2pt](30,15)(40,20)(40,40)(20,20)
\psline(40,40)(24,0)
\psline[linestyle=dotted](24,0)(20,-10)
\psline[linestyle=dotted](0,0)(20,-10)(40,0)
\psline(40,0)(40,20)(20,20)(0,0)(30,15)(60,0)
\psline[linewidth=2pt](60,0)(0,0)
\uput[l](0,0){\gr A}
\uput[ul](24,0){\gr B}
\uput[dr](40,0){\gr G}
\uput[d](60,0){\gr D}
\uput[u](40,40){\gr E}
\uput[ul](20,20){\gr Z}
\uput[r](40,20){\gr H}
\uput[-70](30,15){\gr J}
\uput[d](20,-10){\gr K}
\end{pspicture}
\end{figure}

\begin{proof}
\gr{EB} uzat\i lm\i\c s olsun,
ve \gr A'dan \gr{DZ} do\u grusuna paralel olan \gr{AK}
\echthw\ olsun,
ve \gr{GK} birle\c stirilmi\c s olsun.
\Oun
\begin{gather*}
\grm{GB}:\grm{BA}::\grm{AB}:\grm{BD},\\
\grm{AB}:\grm{BD}::\grm{KB}:\grm{BJ}
\end{gather*}
oldu\u gundan,
ayr\i ca
\begin{equation*}
\grm{GB}:\grm{BA}::\grm{KB}:\grm{BJ}.
\end{equation*}
\Ara\ 
\begin{equation*}
\text{\gr{AJ} \gr{KG}'ya paraleldir.}
\end{equation*}
\Oun\ \palin
\begin{equation*}
\grm{AZ}:\grm{ZE}::\grm{GH}:\grm{HE}
\end{equation*}
(\gar\ her oran $\grm{KJ}:\grm{JE}$ oran\i yla ayn\i d\i r).
\Hoste
\begin{equation*}
\text{\gr{ZH} \gr{AG}'ya paraleldir.}\qedhere
\end{equation*}
\end{proof} 

%\nextlem

\addchap{Lemma VIII (\"Onerme 134)}

\begin{theorem}
Diyagram (\gr{<o bwm'iskos} ``k\"u\c c\"uk sunak'')
\gr{ABGDEZH} olsun,
ve 
\begin{equation*}
\text{\gr{DE} \gr{BG}'ya,
ve \gr{EH} \gr{BZ}'ya, paralel olsun.}
\end{equation*}
\Hoti\
\begin{equation*}
\text{\gr{DZ} da \gr{GH}'ya paraleldir.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=2cm}
\begin{pspicture}(4,4)
\pspolygon(0,0)(4,0)(3,3)(3,1)(1,1)(0,4)
\psset{linestyle=dotted}
\psline(0,0)(3,1)
\psline(4,0)(1,1)
\psline(0,4)(3,3)
\psset{linewidth=2pt,linestyle=solid}
\psline(0,0)(3,3)
\psline(4,0)(0,4)
\uput[d](2,2){\gr A}
\uput[l](0,0){\gr B}
\uput[r](4,0){\gr G}
\uput[160](1,1){\gr D}
\uput[20](3,1){\gr E}
\uput[l](0,4){\gr Z}
\uput[ur](3,3){\gr H}
\end{pspicture}
\end{figure}

\begin{proof}
\gr{BE}, \gr{DG}, ve \gr{ZH}
birle\c stirmi\c s olsun.
\Ara
\begin{equation*}
\text{\gr{DBE} \"u\c cgeni
\gr{DGE} \"u\c cgenine e\c sittir.}
\end{equation*}
Ortak \gr{DAE} \"u\c cgeni
eklenmi\c s olsun.
\Ara
\begin{equation*}
\text{\gr{ABE} \"u\c cgeninin t\"um\"u,
\gr{GDA} \"u\c cgeninin t\"um\"une e\c sittir.}
\end{equation*}
\Palin\
\gr{BZ} \gr{EH}'ya paralel oldu\u gundan
\begin{equation*}
\text{\gr{BZE} \"u\c cgeni
\gr{ABZ} \"u\c cgenine e\c sittir.}
\end{equation*}
Ortak \gr{ABZ} \"u\c cgeni ayr\i lm\i\c s olsun.
\Ara
\begin{equation*}
\text{kalan \gr{ABE} \"u\c cgeni,
kalan \gr{AHZ} \"u\c cgenine e\c sittir.}
\end{equation*}
\Alla
\begin{equation*}
\text{ \gr{ABE} \"u\c cgeni
\gr{AGD} \"u\c cgenine e\c sittir.}
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{AGD} \"u\c cgeni de,
\gr{AHZ} \"u\c cgenine e\c sittir.}
\end{equation*}
Ortak \gr{AGH} \"u\c cgeni
eklenmi\c s olsun.
\Ara
\begin{equation*}
\text{\gr{GDH} \"u\c cgeninin t\"um\"u,
\gr{GZH} \"u\c cgeninin t\"um\"une e\c sittir.}
\end{equation*}
Ve ayn\i\ \gr{GH} taban\i ndad\i rlar.
\Ara
\begin{equation*}
\text{\gr{GH} \gr{DZ}'ya paraleldir.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma IX (\"Onerme 135)}

\begin{theorem}
\"U\c cgen \gr{ABG} olsun,
ve orada \gr{AD} ve \gr{AE} \diechthw\ olsun,
ve \gr{BG}'ya paralel olan \gr{ZH} \echthw\ olsun,
ve \gr{ZJH} e\u gilmi\c s olsun,
ve
\begin{equation*}
\grm{BJ}:\grm{JG}::\grm{DJ}:\grm{JE}
\end{equation*}
olsun.
\Hoti
\begin{equation*}
\text{\gr{KL} \gr{BG}'ya paraleldir.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=4mm}
\begin{pspicture}(0,-1.25)(24,13.25)
\pspolygon(0,0)(24,0)(12,12)
\pspolygon(12,0)(18,6)(6,6)
\psline(6,0)(12,12)(18,0)
\psline(8,4)(16,4)
\uput[u](12,12){\gr A}
\uput[d](0,0){\gr B}
\uput[d](24,0){\gr G}
\uput[d](6,0){\gr D}
\uput[d](18,0){\gr E}
\uput[ul](6,6){\gr Z}
\uput[ur](18,6){\gr H}
\uput[d](12,0){\gr J}
\uput[l](8,4){\gr K}
\uput[r](16,4){\gr L}
\uput[ul](9,6){\gr M}
\uput[ur](15,6){\gr N}
\end{pspicture}
\end{figure}

\begin{proof}
\Gar
\begin{equation*}
\grm{BJ}:\grm{JG}::\grm{DJ}:\grm{JE}
\end{equation*}
oldu\u gundan,
\ara
\begin{equation*}
\text{kalan BD}:\text{kalan GE}::\grm{DJ}:\grm{JE}.
\end{equation*}
Ve
\begin{equation*}
\grm{BD}:\grm{EG}::\grm{ZM}:\grm{NH};
\end{equation*}
\ara\ ayr\i ca
\begin{equation*}
\grm{ZM}:\grm{NH}::\grm{DJ}:\grm{JE}.
\end{equation*}
\Enallax
\begin{equation*}
\grm{ZM}:\grm{DJ}::\grm{NH}:\grm{JE}.
\end{equation*}
\Alla\ paralellerden
\begin{gather*}
\grm{ZM}:\grm{DJ}::\grm{ZK}:\grm{KJ},\\
\grm{HN}:\grm{JE}::\grm{HL}:\grm{LJ},
\end{gather*}
ve \ara
\begin{equation*}
\grm{ZK}:\grm{KJ}::\grm{HL}:\grm{LJ}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{KL} \gr{HZ}'ya paraleldir,}
\end{equation*}
\hoste\ \gr{GB}'ya.
\end{proof}


%\nextlem

\addchap{Lemma X (\"Onerme 136)}

\begin{theorem}
\.Iki do\u gru \gr{BAE} ve \gr{DAH} \"uzerine,
\gr J noktas\i ndan ge\c cen iki do\u gru \gr{DJ} ve \gr{JE} \diechthw\ olsun, ve
\begin{equation*}
\grm{DJ}\cdot\grm{BG}:\grm{DG}\cdot\grm{BJ}::\grm{JH}\cdot\grm{ZE}:\grm{JE}\cdot\grm{ZH}
\end{equation*}
olsun.
\Hoti
\begin{equation*}
\text{\gr G, \gr A, ve \gr Z'dan ge\c cen [\c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=7mm}
\begin{pspicture}(0,-0.7)(12,8.7)
\pspolygon[linestyle=dotted](0,6)(12,6)(8,8)(6,6)(4,8)(0,6)
\psline[linestyle=dotted](12,6)(8,4)
\psline[linestyle=dotted](0,6)(4,4)
\psline[linestyle=dashed](3,3)(9,3)
\psline(0,0)(8,4)
\psline(12,0)(4,4)
\psline[linewidth=2pt](0,0)(6,6)(12,0)
\psdot(6,3)
\uput[d](6,3){\gr A}
\uput[u](4,4){\gr B}
\uput[ul](3,3){\gr G}
\uput[dl](0,0){\gr D}
\uput[dr](12,0){\gr E}
\uput[ur](9,3){\gr Z}
\uput[u](8,4){\gr H}
\uput[u](6,6){\gr J}
\uput[r](12,6){\gr K}
\uput[l](0,6){\gr L}
\uput[u](4,8){\gr M}
\uput[u](8,8){\gr N}
\end{pspicture}
%\caption{Lemma X}
\end{figure}

\begin{proof}\mbox{}
\begin{itemize}
\item
\gr J'dan ve \gr{GA}'ya paralel olan \gr{KL}
\echthw\ olsun,
\item
\gr{AB} ve \gr{AD} ile 
\gr K ve \gr L noktalar\i nda kesi\c smi\c s olsun,
\item
\gr L'dan \gr{AD}'ya paralel olan \gr{LM} \echthw\ olsun,
\item
\gr{EJ} \gr M'ye uzat\i lm\i\c s olsun,
\item
\gr K'dan \gr{AB}'ya paralel olan \gr{KN} \echthw\ olsun,
\item
\gr{DJ} \gr N'ye uzat\i lm\i\c s olsun.
\end{itemize}
\Oun\ paralellerden
\begin{equation*}
\grm{DJ}:\grm{JN}::\grm{DG}:\grm{GB}
\end{equation*}
olmu\c s oldu\u gundan,
\ara
\begin{equation*}
\grm{DJ}\cdot\grm{GB}=\grm{DG}\cdot\grm{JN}.
\end{equation*}
\Allode{DG}{BJ}.
\Ara
\begin{align*}
	\grm{DJ}\cdot\grm{BG}:\grm{DG}\cdot\grm{BJ}
	&::\grm{GD}\cdot\grm{JN}:\grm{DG}\cdot\grm{BJ}\\
	&::\grm{JN}:\grm{JB}.
\end{align*}
\Alla
\begin{equation*}
\grm{JD}\cdot\grm{BG}:\grm{DG}\cdot\grm{BJ}::\grm{JH}\cdot\grm{ZE}:\grm{JE}\cdot\grm{ZH}
\end{equation*}
varsay\i l\i r,
ve
\begin{align*}
	\grm{JN}:\grm{JB}
	&::\grm{KJ}:\grm{JL}&&\\
	&::\grm{HJ}:\grm{JM}&&\text{(paralellerden)}\\
	&::\grm{JH}\cdot\grm{ZE}:\grm{JM}\cdot\grm{ZE}.&&
\end{align*}
Ve \ara
\begin{equation*}
\grm{JH}\cdot\grm{ZE}:\grm{JE}\cdot\grm{ZH}::\grm{JH}\cdot\grm{ZE}:\grm{JM}\cdot\grm{ZE}.
\end{equation*}
\Ara
\begin{equation*}
\grm{JE}\cdot\grm{ZH}=\grm{JM}\cdot\grm{ZE}.
\end{equation*}
Ve \ara
\begin{equation*}
\grm{JM}:\grm{JE}::\grm{HZ}:\grm{ZE}.
\end{equation*}
\Synthenti\ ve \enallax
\begin{equation*}
\grm{ME}:\grm{EH}::\grm{JE}:\grm{EZ}.
\end{equation*}
\Alla
\begin{equation*}
\grm{ME}:\grm{EH}::\grm{LE}:\grm{EA},
\end{equation*}
ve \ara
\begin{equation*}
\grm{LE}:\grm{EA}::\grm{JE}:\grm{EZ}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{AZ}, \gr{KL}'ya paraleldir.}
\end{equation*}
\Alla
\begin{equation*}
\text{\gr{GA} da [\gr{KL}'ya paraleldir].}
\end{equation*}
\Ara
\begin{equation*}
\text{\gr{GAZ} do\u grudur.}\qedhere
\end{equation*}
\end{proof}

Bunun durumlar\i,
tersi olan
\"onceden yaz\i lm\i\c slar\i nki 
[yani Lem\-ma III'\"un durumlar\i]
gibidir.

\FloatBarrier
\begin{figure}
\centering
\psset{unit=7mm}
\begin{pspicture}(0,-0.7)(12,8.7)
\pspolygon[linestyle=dotted](0,6)(12,6)(8,8)(6,6)(4,8)(0,6)
\psline[linestyle=dotted](12,6)(8,4)
\psline[linestyle=dotted](0,6)(4,4)
\psline[linestyle=dashed](3,3)(9,3)
\psline(0,0)(8,4)
\psline(12,0)(4,4)
\psline[linewidth=2pt](0,0)(6,6)(12,0)
\psdot(6,3)
\uput[d](6,3){\gr A}
\uput[u](4,4){\gr B}
\uput[ul](3,3){\gr G}
\uput[dl](0,0){\gr D}
\uput[dr](12,0){\gr E}
\uput[ur](9,3){\gr Z}
\uput[u](8,4){\gr H}
\uput[u](6,6){\gr J}
\uput[r](12,6){\gr K}
\uput[l](0,6){\gr L}
\uput[u](4,8){\gr M}
\uput[u](8,8){\gr N}
\end{pspicture}
%\caption{Lemma X}
\end{figure}

%\nextlem

\addchap{Lemma XI (\"Onerme 137)}

\begin{theorem}
\"U\c cgen \gr{ABG} [olsun],
ve \gr{AD} \gr{BG}'ya paralel [olsun],
ve \diechthw\ olan \gr{DE},
\gr{BG} ile \gr E noktada kesi\c smi\c s olsun.
\Hoti
\begin{equation*}
\grm{DE}\cdot\grm{ZH}:\grm{EZ}\cdot\grm{HD}::\grm{GB}:\grm{BE}.
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=0.8mm}
\begin{pspicture}(-45,-46)(75,66)
\psline[linewidth=2pt](0,0)(30,60)(75,60)
\psline[linewidth=2pt](30,60)(60,0)
\psdot(-45,0)
\psline(75,60)(-45,0)(60,0)
\psline[linestyle=dotted](0,0)(-20,-40)(60,0)
%\pspolygon(30,60)(-20,-40)(60,0)
%\psline(30,60)(75,60)(-45,0)(60,0)
\uput[u](30,60){\gr A}
\uput[dr](0,0){\gr B}
\uput[dr](60,0){\gr G}
\uput[u](75,60){\gr D}
\uput[d](-45,0){\gr E}
\uput[ul](15,30){\gr Z}
\uput[-25](39,42){\gr H}
\uput[d](-20,-40){\gr J}
\end{pspicture}
%\caption{Lemma XI}
\end{figure}

\begin{proof}
\gr G'dan \gr{AE}'a paralel olan \gr{GJ} \echthw\ olsun,
ve \gr{AB} \gr J'ya uzat\i lm\i\c s olsun.
\Oun
\begin{gather*}
	\grm{GA}:\grm{AH}::\grm{GJ}:\grm{ZH},\\
	\grm{GA}:\grm{AH}::\grm{ED}:\grm{DH}
\end{gather*}
oldu\u gundan,
dahi
\begin{equation*}
	\grm{ED}:\grm{DH}::\grm{JG}:\grm{ZH}.
\end{equation*}
\Ara
\begin{equation*}
\grm{GJ}\cdot\grm{DH}=\grm{ED}\cdot\grm{ZH}.
\end{equation*}
\Allode{EZ}{HD}.
\Ara
\begin{align*}
	\grm{DE}\cdot\grm{ZH}:\grm{DH}\cdot\grm{EZ}
	::&\grm{GJ}\cdot\grm{DH}:\grm{DH}\cdot\grm{EZ}\\
	::&\grm{GJ}:\grm{EZ}\\
	::&\grm{GB}:\grm{BE}.
\end{align*}
\Oun
\begin{equation*}
\grm{DE}\cdot\grm{ZH}:\grm{EZ}\cdot\grm{HD}::\grm{GB}:\grm{BE}.
\end{equation*}
E\u ger \gr{AD} paraleli di\u ger tarafa da \echthw\ ise,
ve \gr{DE} \gr D'dan \gr G'n\i n \"otesine \diechthw\ ise,
ayn\i\ \c sey [do\u grudur].
\end{proof}

%\nextlem

\addchap{Lemma XII (\"Onerme 138)}

\begin{theorem}
Bunlar \c simdi kan\i tlanm\i\c s olunca,
e\u ger \gr{AB} ve \gr{GD} paralel ise,
ve
bunlar\i n \"uzerine 
baz\i\ do\u grular \gr{AD}, \gr{AZ}, \gr{BG}, ve \gr{BZ}
d\"u\c serse,
ve \gr{ED} ve \gr{EG} birle\c stirilirse,
o zaman 
\begin{equation*}
\text{\gr H, \gr M, ve \gr K'dan ge\c cen [\c cizgi'nin] do\u gru oldu\u gu}
\end{equation*}
g\"osterilecek.
\end{theorem}

\begin{figure}
\centering
\psset{unit=8mm}
\begin{pspicture}(0,-0.5)(12,12.5)
\psline[linewidth=2pt](0,0)(12,0)
\psline[linewidth=2pt](0,12)(12,12)
\pspolygon(0,0)(6,12)(12,0)(0,12)(6,0)(12,12)
%\psline(0,0)(6,12)(12,0)
%\psline(0,12)(6,0)(12,12)
\psline[linestyle=dotted](3,6)(9,6)
\uput[u](0,12){\gr A}
\uput[u](12,12){\gr B}
\uput[d](0,0){\gr G}
\uput[d](12,0){\gr D}
\uput[u](6,12){\gr E}
\uput[d](6,0){\gr Z}
\uput[l](3,6){\gr H}
\uput[100](4,8){\gr J}
\uput[r](9,6){\gr K}
\uput[80](8,8){\gr L}
\uput[d](6,6){\gr M}
\end{pspicture}
%\caption{Lemma XII}
\end{figure}

\begin{proof}
\Gar\ \gr{DAZ}
\"u\c cgen oldu\u gundan,
ve \gr{AE} \gr{DZ}'ya paralel oldu\u gundan,
ve
\gr{DZ}'ya \gr G'da d\"u\c sen
\gr{EG} \diechthw\ oldu\u gundan,
\"onceden yaz\i lm\i\c slara g\"ore
\begin{equation*}
\grm{DZ}:\grm{ZG}::\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}
\end{equation*}
meydana gelir.
\Palin\ \gr{GBZ}
\"u\c cgen oldu\u gundan,
ve \gr{GD}'ya paralel olan
\gr{BE} \echthw\ oldu\u gundan,
ve 
\gr{GZD}'ya \gr D'da d\"u\c sen \gr{DE}
\diechthw\ oldu\u gundan
\begin{equation*}
\grm{GZ}:\grm{ZD}::\grm{DE}\cdot\grm{LK}:\grm{DK}\cdot\grm{LE}
\end{equation*}
meydana gelir.
\Ara\ \anapalin
\begin{equation*}
\grm{DZ}:\grm{ZG}::\grm{DK}\cdot\grm{LE}:\grm{DE}\cdot\grm{LK}
\end{equation*}
meydana gelir.
\begin{equation*}
\grm{DZ}:\grm{ZG}::\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}
\end{equation*}
da oldu.
Ve \ara
\begin{equation*}
\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}
::\grm{DK}\cdot\grm{LE}:\grm{DE}\cdot\grm{KL}.
\end{equation*}
\Oun\ 
iki do\u gru \gr{EG} ve \gr{ED},
iki do\u gru \gr{GML} ve \gr{DMJ}'ya
\diechthw\ oldu\u gundan,
ve
\begin{equation*}
\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}::\grm{DK}\cdot\grm{EL}:\grm{DE}\cdot\grm{LK}
\end{equation*}
oldu\u gundan,
\ara
\begin{equation*}
\text{\gr H, \gr M, ve \gr K'dan [ge\c cen \c cizgi]
do\u grudur;}
\end{equation*}
\gar\ bu g\"osterilmi\c s oldu.
\end{proof}

%\nextlem

\addchap{Lemma XIII (\"Onerme 139)}

\begin{theorem}
\Alla\ \dee\
\gr{AB} ve \gr{GD} paralel
olmas\i n,
\alla\ \gr N'de kesi\c smi\c s olsun.
\Hoti\ \palin\ 
\begin{equation*}
\text{\gr H, \gr M, ve \gr K'dan ge\c cen [\c cizgi]
do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=0.25mm}
\begin{pspicture}(0,-20)(360,200)
\psline[linewidth=2pt](360,0)(0,0)(180,180)
\pspolygon(180,180)(180,0)(60,60)(360,0)(120,120)(120,0)
\psline[linestyle=dotted](120,30)(180,90)
\uput[ul](60,60){\gr A}
\uput[u](180,180){\gr B}
\uput[d](120,0){\gr G}
\uput[d](360,0){\gr D}
\uput[ul](120,120){\gr E}
\uput[d](180,0){\gr Z}
\uput[130](120,48){\gr J}
\uput[dl](120,30){\gr H}
\uput[ur](180,90){\gr K}
\uput[120](155,105){\gr L}
\uput[-60](135,45){\gr M}
\uput[d](0,0){\gr N}
\end{pspicture}
%\caption{Lemma XIII}
\end{figure}

\begin{proof}
\"U\c c do\u gru \gr{AN}, \gr{AZ}, ve \gr{AD} \"uzerine
ayn\i\ \gr G noktas\i ndan
iki do\u gru \gr{GE} ve \gr{GD}
\diechthw\ oldu\u gundan,
\begin{equation*}
\grm{GE}\cdot\grm{HJ}:\grm{GH}:\grm{JE}::\grm{GN}\cdot\grm{ZD}:\grm{ND}\cdot\grm{GZ}
\end{equation*}
meydana gelir.
\Palin\
ayn\i\ \gr D noktas\i ndan,
\"u\c c do\u gru \gr{BN}, \gr{BG}, ve \gr{BZ} \"uzerine
iki do\u gru \gr{DE} ve \gr{DN}
\diechthw\ oldu\u gundan
\begin{equation*}
\grm{NG}\cdot\grm{ZD}:\grm{ND}:\grm{ZG}::\grm{DK}\cdot\grm{EL}:\grm{DE}\cdot\grm{KL}.
\end{equation*}
\Alla
\begin{equation*}
\grm{NG}\cdot\grm{ZD}:\grm{ND}\cdot\grm{GZ}::\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}
\end{equation*}
g\"osterilmi\c s oldu.
Ve \oun
\begin{equation*}
\grm{GE}\cdot\grm{JH}:\grm{GH}\cdot\grm{JE}::\grm{DK}\cdot\grm{EL}:\grm{DE}\cdot\grm{KL}.
\end{equation*}
\Dee\ \"onceden yaz\i lm\i\c slara g\"ore
\begin{equation*}
\text{\gr H, \gr M, ve \gr K'dan ge\c cen [\c cizgi]
do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma XIV (\"Onerme 140)}

\begin{theorem}
\gr{AB} \gr{GD}'ya paralel olsun,
ve \gr{AE} ve \gr{GB} \diechthw\ olsun,
ve [\"oyle] bir nokta \gr{BH}'da \gr Z [olsun] ki
\begin{equation*}
\grm{DE}:\grm{EG}::\grm{GB}\cdot\grm{HZ}:\grm{ZB}\cdot\grm{GH}
\end{equation*}
[olsun].
\Hoti
\begin{equation*}
\text{\gr A, \gr Z, ve \gr D'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=5mm}
\begin{pspicture}(-4,-5)(12,13)
\psline[linewidth=2pt](6,0)(0,12)(12,12)
\psline[linewidth=2pt](6,6)(0,12)
\psdot(0,0)
\psline(12,12)(0,0)(12,0)
\psline[linestyle=dashed,linewidth=2pt](6,6)(12,0)
\psline[linestyle=dotted](12,0)(8,-4)(-4,-4)(0,0)
\psline[linestyle=dotted](8,-4)(6,0)
\uput[ul](0,12){\gr A}
\uput[ur](12,12){\gr B}
\uput[ul](0,0){\gr G}
\uput[r](12,0){\gr D}
\uput[dl](6,0){\gr E}
\uput[r](6,6){\gr Z}
\uput[l](4,4){\gr H}
\uput[dr](8,-4){\gr J}
\uput[d](-4,-4){\gr K}
\end{pspicture}
\end{figure}

\begin{proof}\mbox{}
\begin{itemize}
\item
\gr D'dan \gr{BG}'ya paralel olan \gr{DJ}
\echthw\ olsun,
ve \gr{AE} \gr J'ya uzat\i lm\i\c s olsun,
\item
\gr J'dan \gr{GD}'ya paralel olan \gr{JK}
[\echthw\ olsun],
ve \gr{BG} \gr K'ya uzat\i lm\i\c s olsun.
\end{itemize}
\Oun
\begin{gather*}
\grm{DE}:\grm{EG}::\grm{GB}\cdot\grm{ZH}:\grm{ZB}\cdot\grm{GH},\\
\begin{aligned}
	\grm{DE}:\grm{EG}
  &::\grm{DJ}:\grm{GH}\\
	&::\grm{DJ}\cdot\grm{BZ}:\grm{GH}\cdot\grm{BZ}
\end{aligned}
\end{gather*}
oldu\u gundan, \ara
\begin{equation*}
\grm{BG}\cdot\grm{ZH}::\grm{DJ}\cdot\grm{BZ}.
\end{equation*}
\Ara
\begin{align*}
	\grm{GB}:\grm{BZ}
	&::\grm{DJ}:\grm{HZ}\\
	&::\grm{GK}:\grm{HZ}
\end{align*}
orant\i\ vard\i r.
\Ara\ ayr\i ca
\begin{align*}
	\text{\gr{KB} t\"um\"u}:\text{\gr{BH} t\"um\"u}
	&::\grm{KG}:\grm{ZH}\\
	&::\grm{DJ}:\grm{ZH}.
\end{align*}
\Alla\ paralellerden
\begin{equation*}
\grm{KB}:\grm{BH}::
\left\{
\begin{gathered}
	\grm{JA}:\grm{AH},\\
	\grm{DJ}:\grm{ZH}.
\end{gathered}\right.
\end{equation*}
Ve \gr{DJ} ve \gr{ZH} paraleldir.
\Ara
\begin{equation*}
\text{\gr A, \gr Z, ve \gr D noktalar\i ndan [ge\c cen \c cizgi] do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma XV (\"Onerme 141)}

\begin{theorem}
Bu \"onceden bak\i lm\i\c s olunca,
\gr{AB} \gr{GD}'ya paralel olsun,
ve bunlar\i n \"uzerine
do\u grular \gr{AZ}, \gr{ZB}, \gr{GE}, ve \gr{ED}
d\"u\c sm\"u\c s olsun,
ve \gr{BG} ve \gr{HK} birle\c stirilmi\c s olsun.
\Hoti
\begin{equation*}
\text{\gr A, \gr M, ve \gr D'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=8mm}
\begin{pspicture}(0,-0.5)(12,12.5)
\psline[linewidth=2pt](0,0)(12,0)
\psline[linewidth=2pt](0,12)(12,12)
\psline(0,12)(6,0)(12,12)(0,0)(6,12)(12,0)
\psline[linestyle=dashed](3,6)(9,6)
\psline[linestyle=dotted](12,0)(0,12)
\uput[u](0,12){\gr A}
\uput[u](12,12){\gr B}
\uput[d](0,0){\gr G}
\uput[d](12,0){\gr D}
\uput[u](6,12){\gr E}
\uput[d](6,0){\gr Z}
\uput[l](3,6){\gr H}
\uput[100](4,8){\gr J}
\uput[r](9,6){\gr K}
\uput[80](8,8){\gr L}
\uput[d](6,6){\gr M}
\end{pspicture}
%\caption{Lemma XV}
\end{figure}

\begin{proof}
\gr{DM} birle\c stirilmi\c s olsun 
ve \gr J'ya uzat\i lm\i\c s olsun.
\Oun\
\gr{BGZ} \"u\c cgeninin \gr B tepe noktas\i ndan \gr{GD}'ya paralel olan 
\gr{BE} \echthw\ oldu\u gundan
ve \gr{DE} \diechthw\ oldu\u gundan,
\begin{equation*}
\grm{GZ}:\grm{ZD}::\grm{DE}\cdot\grm{KL}:\grm{EL}\cdot\grm{KD}
\end{equation*}
meydana gelir.
Ayr\i ca
\begin{equation*}
\grm{DE}\cdot\grm{KL}:\grm{DK}\cdot\grm{LE}
::\grm{GH}\cdot\grm{JE}:\grm{GE}\cdot\grm{HJ}
\end{equation*}
(\gar\ \"u\c c do\u gru \gr{GL}, \gr{DJ}, ve \gr{HK} \"uzerine
ayn\i\ \gr E noktas\i ndan iki do\u gru \gr{EG} ve \gr{ED}
\diechthw t\"ur).
\Ara
\begin{equation*}
\grm{DZ}:\grm{ZG}::\grm{GE}\cdot\grm{HJ}:\grm{GH}\cdot\grm{JE}.
\end{equation*}
Ayr\i ca
\begin{equation*}
\text{\gr J, \gr M, ve \gr D'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\Ara\ \"onceden yaz\i lm\i\c stan ayr\i ca
\begin{equation*}
\text{\gr A, \gr M, ve \gr D'dan [ge\c cen \c cizgi] do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma XVI (\"Onerme 142)}

\begin{theorem}
\.Iki do\u gru \gr{AB} ve \gr{AG} \"uzerine
ayn\i\ \gr A noktas\i ndan \gr{DB} ve \gr{DE} \diechthw\ olsun,
ve bunlarda \gr H ve \gr J noktalar\i\ al\i nm\i\c s olsun,
ve
\begin{equation*}
\grm{EH}\cdot\grm{ZD}:\grm{DE}\cdot\grm{HZ}::\grm{BJ}\cdot\grm{GD}:\grm{BD}\cdot\grm{GJ}
\end{equation*}
olsun.
\Hoti
\begin{equation*}
\text{\gr A, \gr H, ve \gr J'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
%\psset{unit=8mm}
\begin{pspicture}(0,-0.5)(10,8.5)
\psline[linewidth=2pt](0,0)(8,8)(8,0)
\psline(0,0)(10,0)(5,5)
\psline[linestyle=dotted,linewidth=2pt](4,0)(8,8)
\psline[linestyle=dotted](4,4)(8,4)
\psdot(10,0)
\uput[u](8,8){\gr A}
\uput[d](0,0){\gr B}
\uput[d](8,0){\gr G}
\uput[d](10,0){\gr D}
\uput[ul](5,5){\gr E}
\uput[dl](8,2){\gr Z}
\uput[d](6,4){\gr H}
\uput[d](4,0){\gr J}
\uput[ul](4,4){\gr K}
\uput[r](8,4){\gr L}
\end{pspicture}
\end{figure}

\begin{proof}
\gr H'dan \gr{BD}'ya paralel olan \gr{KL} \echthw\ olsun.
\Oun
\begin{equation*}
\grm{EH}\cdot\grm{ZD}:\grm{DE}\cdot\grm{ZH}::\grm{BJ}\cdot\grm{GD}:\grm{BD}\cdot\grm{GJ}
\end{equation*}
oldu\u gundan, \alla
\begin{align*}
	\grm{EH}\cdot\grm{ZD}:\grm{DE}\cdot\grm{HZ}
	&::\grm{HE}:\grm{ED}\And\grm{DZ}:\grm{ZH}\\
	&::\grm{KH}:\grm{BD}\And\grm{GD}:\grm{HL}
\end{align*}
oldu\u gundan,
ve ayr\i ca
\begin{equation*}
\grm{BJ}\cdot\grm{GD}:\grm{BD}\cdot\grm{GJ}
	 ::\grm{JB}:\grm{BD}\And\grm{DG}:\grm{GJ}
\end{equation*}
oldu\u gundan, \ara
\begin{equation*}
\grm{KH}:\grm{BD}\And\grm{GD}:\grm{HL}::\grm{BJ}:\grm{BD}\And\grm{DG}:\grm{GJ}.
\end{equation*}
Ve
\begin{equation*}
\grm{KH}:\grm{BD}::\grm{KH}:\grm{BJ}\And\grm{BJ}:\grm{BD};
\end{equation*}
\ara
\begin{equation*}
\grm{KH}:\grm{BJ}\And\grm{BJ}:\grm{BD}\And\grm{GD}:\grm{HL}
::\grm{BJ}:\grm{BD}\And\grm{DG}:\grm{GJ}.
\end{equation*}
Ortak $\grm{BJ}:\grm{BD}$ oran\i\ kovulmu\c s olsun.
\Ara\ kalan
\begin{align*}
\grm{KH}:\grm{BJ}\And\grm{GD}:\grm{HL}
&::\grm{DG}:\grm{GJ}\\
&::\grm{DG}:\grm{HL}\And\grm{HL}:\grm{JG}.
\end{align*}
Ve \palin\ ortak $\grm{DG}:\grm{HL}$ oran\i\ kovulmu\c s olsun.
\Ara\ kalan
\begin{equation*}
\grm{KH}:\grm{BJ}
::\grm{HL}:\grm{JG}.
\end{equation*}
\Enallax
\begin{equation*}
\grm{KH}:\grm{HL}
::\grm{BJ}:\grm{JG},
\end{equation*}
ve \gr{KL} ve \gr{BG} paraleldir.
\Ara
\begin{equation*}
\text{\gr A, \gr H, ve \gr J noktalar\i ndan [ge\c cen \c cizgi] do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma XVII (\"Onerme 143)}

\begin{theorem}
\Alla\ \dee\ \gr{AB} ve \gr{GD} paralel olmas\i n,
\alla\ \gr N'de kesi\c smi\c s olsun.
\end{theorem}

\begin{figure}
\centering
\psset{unit=0.25mm}
\begin{pspicture}(0,-20)(360,200)
\psline[linewidth=2pt](360,0)(0,0)(180,180)
\psline(360,0)(120,120)(120,0)(180,180)(180,0)(60,60)
\psline[linestyle=dashed](120,30)(180,90)
\psline[linestyle=dotted](360,0)(60,60)
\uput[ul](60,60){\gr A}
\uput[u](180,180){\gr B}
\uput[d](120,0){\gr G}
\uput[d](360,0){\gr D}
\uput[ul](120,120){\gr E}
\uput[d](180,0){\gr Z}
\uput[130](120,48){\gr J}
\uput[dl](120,30){\gr H}
\uput[ur](180,90){\gr K}
\uput[120](155,105){\gr L}
\uput[-60](135,45){\gr M}
\uput[d](0,0){\gr N}
\end{pspicture}
%\caption{Lemma XIII}
\end{figure}

\begin{proof}
\Oun\
ayn\i\ \gr A noktas\i ndan
\"u\c c do\u gru \gr{BN}, \gr{BG}, ve \gr{BZ} \"uzerine
iki do\u gru \gr{DE} ve \gr{DN} \diechthw\ oldu\u gundan,
\begin{equation*}
\grm{ND}\cdot\grm{GZ}:\grm{NG}\cdot\grm{DZ}::\grm{DE}\cdot\grm{KL}:\grm{EL}\cdot\grm{KD}.
\end{equation*}
Ve
\begin{equation*}
\grm{ED}\cdot\grm{KL}:\grm{EL}\cdot\grm{KD}::\grm{EJ}\cdot\grm{GH}:\grm{EG}\cdot\grm{JH}
\end{equation*}
(\gar\ \palin\ 
\"u\c c do\u gru \gr{GL}, \gr{DJ}, ve \gr{HK} \"uzerine
ayn\i\ \gr E noktas\i ndan
\diechthw t\"ur
iki do\u gru \gr{EG} ve \gr{ED}).
Ayr\i ca
\begin{equation*}
\grm{EJ}\cdot\grm{GH}:\grm{EG}\cdot\grm{JH}::\grm{ND}\cdot\grm{GZ}:\grm{NG}\cdot\grm{ZD}.
\end{equation*}
\Dee\ \"onceden yaz\i lm\i\c stan
\begin{equation*}
\text{\gr A, \gr J, ve \gr L'dan [ge\c cen \c cizgi] do\u grudur.}
\end{equation*}
\Ara
\begin{equation*}
\text{\gr A, \gr M, ve \gr L'dan [ge\c cen \c cizgi] do\u grudur.}\qedhere
\end{equation*}
\end{proof}

%\nextlem

\addchap{Lemma XVIII (\"Onerme 144)}

\begin{theorem}
\"U\c cgen \gr{ABG} [olsun],
ve \gr{BG}'ya paralel olan \gr{AD} \echthw\ olsun,
ve \gr{DE} ve \gr{ZH} \diechthw\ olsun.
Ayr\i ca
\begin{equation*}
\grm{EB}^2:\grm{EG}\cdot\grm{GB}::\grm{BH}:\grm{HG}
\end{equation*}
olsun
(\gr{>'estw d`e ws t`o >ap`o EB pr`os t`o <up`o EGB, o<'utws <h BH pr`os t`hn HG}).
\Hoti\ \gr{BD} birle\c stirilirse,
\begin{equation*}
\text{\gr J, \gr K, \gr G'dan [ge\c cen \c cizgi] do\u gru olur.}
\end{equation*}
\end{theorem}

\begin{figure}
\centering
\psset{unit=17mm}
\begin{pspicture}(0,-0.3)(6,3.3)
\pspolygon(3,0)(3,3)(6,3)
\psline(3,3)(6,0)(0,0)(6,3)
\psline(4,0)(4,2)
\psline[linestyle=dotted](6,0)(3,1.5)
\uput[ul](3,3){\gr A}
\uput[d](3,0){\gr B}
\uput[d](6,0){\gr G}
\uput[u](6,3){\gr D}
\uput[d](0,0){\gr E}
\uput[u](4,2){\gr Z}
\uput[d](4,0){\gr H}
\uput[ul](3,1.5){\gr J}
\uput[l](4,1){\gr K}
\end{pspicture}
\end{figure}

\begin{proof}
\begin{math}
\grm{EB}^2:\grm{EG}\cdot\grm{GB}::\grm{BH}:\grm{HG}
\end{math}
oldu\u gundan,
$\grm{EG}\cdot\grm{GB}:\grm{EB}\cdot\grm{BG}$ 
ile ayn\i\ olan
ortak $\grm{GE}:\grm{EB}$ oran\i\ eklenmi\c s olsun.
\Ara\ \diisou
\begin{equation*}
\left.
\begin{gathered}
	\grm{EB}^2:\grm{EB}\cdot\grm{BG}\\
	\grm{EB}:\grm{BG}
\end{gathered}
\right\}
::\grm{BH}:\grm{HG}\And
\underbrace{\grm{EG}\cdot\grm{GB}:\grm{EB}\cdot\grm{BG}}
_{\displaystyle\grm{EG}:\grm{EB}}.
\end{equation*}
\Hoste
\begin{align*}
	\grm{EB}^2:\grm{EB}\cdot\grm{BG}
	&::\grm{BH}:\grm{HG}\And\grm{EG}:\grm{EB}\\
	&::\grm{EG}\cdot\grm{BH}:\grm{EB}\cdot\grm{GH}.
\end{align*}
\"Onceden yaz\i lm\i\c s lemmadan
\begin{equation*}
\grm{EB}:\grm{BG}::\grm{DZ}\cdot\grm{JE}:\grm{DE}\cdot\grm{ZJ},
\end{equation*}
ve \ara
\begin{equation*}
\grm{GE}\cdot\grm{BH}:\grm{GH}\cdot\grm{EB}
::\grm{DZ}\cdot\grm{JE}:\grm{DE}\cdot\grm{ZJ}.
\end{equation*}
\Ara
\begin{equation*}
\text{\gr J, \gr K, ve \gr G'dan [ge\c cen \c cizgi] do\u grudur,}
\end{equation*}
\gar\ bu, [Lemma X'un] kar\c s\i t tersinin durumlar\i ndad\i r.
\end{proof}

%\nextlem

\addchap{Lemma XIX (\"Onerme 145)}

\begin{theorem}
\"U\c c do\u gru \gr{AB}, \gr{AG}, \gr{AD} \"uzerine
bir \gr E noktas\i ndan iki do\u gru \gr{EZ} ve \gr{EB} \diechthw\ olsun.
Ayr\i ca
\begin{equation*}
\grm{EZ}:\grm{ZH}::\grm{JE}:\grm{JH}
\end{equation*}
olsun.
\Hoti
\begin{equation*}
\grm{BE}:\grm{BG}::\grm{ED}:\grm{DG}
\end{equation*}
meydana gelir.
\end{theorem}

\begin{figure}
\centering
\psset{unit=5mm}
\begin{pspicture}(0,-1)(15,11)
\psline[linewidth=2pt](0,0)(10,10)(12,0)
\psline[linewidth=2pt](10,10)(10,0)
\psdot(15,0)
\psline(0,0)(15,0)(7.5,7.5)
\psline[linestyle=dotted](5,5)(11,5)
\uput[u](10,10){\gr A}
\uput[d](0,0){\gr B}
\uput[d](10,0){\gr G}
\uput[d](12,0){\gr D}
\uput[d](15,0){\gr E}
\uput[ul](7.5,7.5){\gr Z}
\uput[dl](10,5){\gr H}
\uput[ur](11.25,3.75){\gr J}
\uput[ul](5,5){\gr K}
\uput[ur](11,5){\gr L}
\end{pspicture}
\end{figure}

\begin{proof}
\gr H'dan \gr{BE}'a paralel olan \gr{LK} \echthw\ olsun.
\Oun
\begin{equation*}
\grm{EZ}:\grm{ZH}::\grm{EJ}:\grm{JH}
\end{equation*}
oldu\u gundan, \alla
\begin{gather*}
\grm{EZ}:\grm{ZH}::\grm{EJ}:\grm{JH},\\
\grm{EJ}:\grm{JH}::\grm{DE}:\grm{HL}
\end{gather*}
oldu\u gundan,
\ara\ ayr\i ca
\begin{equation*}
\grm{BE}:\grm{HK}::\grm{DE}:\grm{HL}.
\end{equation*}
\Enallax
\begin{equation*}
\grm{EB}:\grm{ED}::\grm{KH}:\grm{HL}.
\end{equation*}
Ve
\begin{equation*}
\grm{KH}:\grm{HL}::\grm{BG}:\grm{GD}.
\end{equation*}
\Ara\ ayr\i ca
\begin{equation*}
\grm{BE}:\grm{ED}::\grm{BG}:\grm{GD}.
\end{equation*}
\Enallax
\begin{equation*}
\grm{EB}:\grm{BG}::\grm{ED}:\grm{DG}.\qedhere
\end{equation*}
\end{proof}

[Di\u ger] durumlar da benzerdir.

%\clearpage

%\nextlem

\appendix

\addchap{Ek}

\addsec{Fiiller S\"ozl\"u\u g\"u}

\begin{compactdesc}
\item[\gr{>'agw}] ilerle= (\"orne\u gin Lemma I'de:
\gr{\textbf{>'Hqjw} di`a to~u Z t~h| BD par'al\-lhlos <h ZL},
``\gr Z'dan \gr{BD}'ya paralel olan \gr{ZL} \emph{\echthw\ olsun}'')
  \begin{compactdesc}
	\item[\gr{>ag'agw}] (sadece Lemma IV'te:
	\gr{>E`an g`ar di`a to~u E t~h| JG par'allhlon \textbf{>ag'agw} t`hn E\myXi},
	``E\u ger \gr E'dan \gr{JG}'ya paralel olan \gr{E\myXi} \emph{ilerlersem}'')
  \item[\gr{di'agw}] s\"urd\"ur= (\"orne\u gin Lemma III'te:
	\gr{E>is tre~is e>uje'ias t`as AB GA DA \textbf{di'hqjwsan} d'uo e>uje~iai a<i JE JD},
	``\"U\c c do\u gru \gr{AB}, \gr{GA}, ve \gr{DA} \"uzerine
iki do\u gru \gr{JE} ve \gr{JD} \emph{\diechthw\ olsun}'')
  \end{compactdesc}
\item[\gr{<air'ew}] \mbox{}
  \begin{compactdesc}
  \item[\gr{>afair'ew}] ay\i r= (Lemma VIII)
  \end{compactdesc}
%\item[\gr{>ait'ew}] rica et=
%\item[\gr{>all'attw}]\mbox{}
%  \begin{compactdesc}
%  \item[\gr{parall'attw}] sap=
%  \end{compactdesc}
%\item[\gr{<'aptw}] \emph{med.}\ dokun=
%\item[\gr{<arm'ozw}]\mbox{}
%  \begin{compactdesc}
%  \item[\gr{>efarm'ozw}] uygula=
%  \end{compactdesc}
\item[\gr{b'allw}]\mbox{}
  \begin{compactdesc}
  \item[\gr{>ekb'allw}] uzat=
	\item[\gr{metab'allw}] de\u gi\c stir=
%  \item[\gr{parab'allw}] uygula=
%  \item[\gr{prosekb'allw}] uzat=
  \end{compactdesc}
\item[\gr{gign'omai}] ol= (do\u g=, meydana gel=)
\item[\gr{gr'afw}] \mbox{}%\c ciz=
  \begin{compactdesc}
%  \item[\gr{>anagr'afw}] \c ciz=
  \item[\gr{progr'afw}] \"onceden yaz=
  \end{compactdesc}
\item[\gr{de'iknumi}] g\"oster=
\begin{compactdesc}
\item[\gr{>apode'iknumi}] kan\i tla= (Lemma XII)
\end{compactdesc}
%\item[\gr{e>il'ew}] (Lemma XVI)
\item[\gr{e>imi}] ol=
\item[\gr{>er~w}] s\"oyle= (\gr{l'egw}'nun gelecek zaman\i\ olarak kullan\i l\i r.
Lemma VI'da sadece:
\gr{<ekat'erwn g`ar t~wn \textbf{e>irhm'enwn} <o a>ut'os >estin t~w| t~hs JH pr`os t`hn HE l'ogos},
``\gar\ \emph{s\"oylenmi\c s} [iki oran\i n] her biri,
$\grm{JH}:\grm{HE}$ oran\i yla ayn\i d\i r'')
\item[\gr{>'eqw}] -i ol=
%  \begin{compactdesc}
%  \item[\gr{peri'eqw}] i\c cer=
%  \end{compactdesc}
\item[\gr{ze'ugnumi}] birle\c stir=
\begin{compactdesc}
\item[\gr{>epize'ugnumi}] birle\c stir=
\end{compactdesc}
\item[\gr{jewr'ew}] \mbox{}
\begin{compactdesc}
\item[\gr{projewr'ew}] \"onceden bak= (Sadece Lemma XV'te:
\gr{To'utou protejewrhm'enou} [\emph{genitivus absolutus}],
``Bu \"onceden bak\i lm\i\c s olunca'')
\end{compactdesc}
%\item[\gr{<'isthmi}] dik=
%  \begin{compactdesc}
%  \item[\gr{di-'isthmi}] (\gr{di'asthma} uzunluk)
%  \item[\gr{>ef'isthmi}] -in \"uzerine dik=
%  \item[\gr{sun'isthmi}] in\c sa et=
%  \end{compactdesc}
%\item[\gr{kal'ew}] \emph{med.} -e den=
\item[\gr{ke~imai}] otur=
  \begin{compactdesc}
%  \item[\gr{>ekke~imai}] oturtul=
  \item[\gr{proske~imai}] eklen=
  \item[\gr{<upoke~imai}] varsay\i l=
  \end{compactdesc}
	\item[\gr{kl'aw}] e\u g= (Lemma IX:
	\gr{\textbf{kekl'asjw} <h ZJH}, ``\gr{ZJH} \emph{e\u gilmi\c s olsun}'')
	\item[\gr{kro'uw}]\mbox{}
	\begin{compactdesc}
	\item[\gr{>ekkro'uw}] kov=
	\end{compactdesc}
\item[\gr{lamb'anw}] al= (Lemma XVI'da:
\gr{>ep' a>ut~wn \textbf{e>il'hfjw} shme~ia t`a H J},
``bunlarda \gr H ve \gr J noktalar\i\ \emph{al\i nm\i\c s olsun}'')
%  \begin{compactdesc}
%  \item[\gr{>apolamb'anw}] ay\i r=
%  \end{compactdesc}
%\item[\gr{l'egw}] (\gr{leg'omenoc} s\"ozde)
%\item[\gr{pera'inw}] s\i n\i rla=
%\item[\gr{perat'ow}] s\i n\i rland\i r=
\item[\gr{p'iptw}]\mbox{}
  \begin{compactdesc}
  \item[\gr{>emp'iptw}] \"uzerine d\"u\c s=
%  \item[\gr{prosp'iptw}] (\emph{acc.}\ ile) \"uzerine d\"u\c s=
  \item[\gr{sump'iptw}] kesi\c s= (her zaman
	\gr{\textbf{sumpipt'etw}} [tekil] veya
	\gr{\textbf{sumpipt'etwsan}} [\c co\u gul],
	``\emph{kesi\c smi\c s olsun}'')
  \end{compactdesc}
%\item[\gr{poi'ew}] yap=
%\item[\gr{te'inw}]\mbox{}
%  \begin{compactdesc}
%  \item[\gr{<upote'inw}] raptet=
%  \end{compactdesc}
%\item[\gr{t'emnw}] kes=
%  \begin{compactdesc}
%  \item[\gr{d'iqa t'emnw}] ikiye b\"ol=
%  \end{compactdesc}
\item[\gr{t'ijhmi}] koy= (Lemma VI:
\gr{>e`an g`ar >ep`i t~hs EB \textbf{j~w} t~h| HB >'ishn t`hn BJ},
``\gar\ e\u ger \gr{EB}'da \gr{HB}'ya e\c sit olan \gr{BJ}'y\i\ \emph{koyarsam}'')
\item[\gr{tugq'anw}] rastla= (\gr{\textbf{tuq'on}}, ``rasgele'': a\c sa\u g\i ya bak\i n)
\end{compactdesc}


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

\begin{description}
\item[\gr{>all'a}] \alla
\item[\gr{>'allo d'e ti tuq`on t`o <up`o t~wn \dots\ \dots}]
\Allode{\dots}{\dots} (\"orne\u gin Lemma III'e bak\i n)
\item[\gr{>'ara}] \ara
%\item[\gr{di'a}] \dia
\item[\gr{di`a ta>ut'a%, di`a t`a a>ut'a
}] \diatauta
\item[\gr{g'ar}] \gar
\item[{[\emph{genitivus absolutus}]}] -ince
\item[\gr{d'e}] de, ve
\item[\gr{d'h}] \dee
\item[\gr{>epe'i}] -di\u ginden
\item[\gr{ka'i}] de, dahi, ve, ayr\i ca---\"orne\u gin Lemma I'de:
  \begin{center}
\gr{>'estin >'ara <ws <h Eb pr`os t`hn BL,\\
o<'utws >en parall'hlw| <h EK pr`os t`hn KZ,\\
\textbf{ka`i} <h EJ pr`os t`hn JH},\\
\mbox{}\\
    \Alla\ paralellerden
\begin{math}
\grm{KB}:\grm{BH}::
\left\{
\begin{gathered}
	\grm{JA}:\grm{AH},\\
	\grm{DJ}:\grm{ZH}.
\end{gathered}\right.
\end{math}
  \end{center}
\item[\gr{kat`a t`a a>ut'a}] \katatauta
\item[\gr{m'en\dots d'e}] $\bullet$\dots$\bullet$\dots de
%\item[\gr{m'hn}] \meen
\item[\gr{o>~un}] \oun
\item[\gr{p'alin}] \palin
%\item[\gr{te\dots ka'i}] hem\dots hem
%\item[\gr{to'inun}] \toinun
\item[\gr{tout'estin}]
yani---\"orne\u gin Lemma I'de:
\begin{center}
\gr{>estin <ws <h DA pr`os t`hn AZ,\\
\textbf{tout'estin} >en parall'hlw| <ws <h BA pr`os t`hn AL,\\
o<'utws <h GA pr`os t`hn AH},\\
\mbox{}\\
$\underbrace{\grm{DA}:\grm{AZ}}_{\displaystyle\grm{BA}:\grm{AL}}
::\grm{GA}:\grm{AH}$,
\end{center}
ve
Lemma III'te:
\begin{center}
\gr{<ws t`o <up`o t~wn EJ HZ pr`os t`o <up`o EZ HJ,}\\
\gr{o<'utws t`o <up`o EZ JM pr`os t`o <up`o EZ HJ,}\\
\gr{\textbf{tout'estin} <h JM pr`os JH,}\\
\gr{\textbf{tout'estin} <h LJ pr`os t`hn JK},\\
\mbox{}\\
$\begin{aligned}
	\grm{EJ}\cdot\grm{HZ}:\grm{EZ}\cdot\grm{HJ}
	&::\grm{EZ}\cdot\grm{JM}:\grm{EZ}\cdot\grm{HJ}\\
	&::\grm{JM}:\grm{JH}\\
	&::\grm{LJ}:\grm{JK}.
\end{aligned}$
\end{center}
\item[\gr{<'wste}] \hoste
\end{description}

\addsec{Yunan Alfabesi}

\begin{center}
\begin{tabular}{ccl}
\gr A&\gr a&alfa\\
\gr B&\gr b&beta\\
\gr G&\gr g&gamma\\
\gr D&\gr d&delta\\
\gr E&\gr e&epsilon (``\c c\i plak e'')\\
\gr Z&\gr z&zeta\\
\gr H&\gr h&eta\\
\gr J&\gr j&theta\\
\gr I&\gr i&iota\\
\gr K&\gr k&kappa\\
\gr L&\gr l&lambda\\
\gr M&\gr m&m\"u
\end{tabular}
\hfill
\begin{tabular}{ccl}
\gr N&\gr n&n\"u\\
\gr{\myXi}&\gr x&ksi\\
\gr O&\gr o&omikron (``k\"u\c c\"uk o'')\\
\gr P&\gr p&pi\\
\gr R&\gr r&rho\\
\gr S&\gr{sv, s}&sigma\\
\gr T&\gr t&tau\\
\gr U&\gr u&\"upsilon\\
\gr F&\gr f&phi\\
\gr Q&\gr q&khi\\
\gr Y&\gr y&psi\\
\gr W&\gr w&omega (``b\"uy\"uk o'')
\end{tabular}
\end{center}
Harfler, Yunan Font Derne\u gi'nin ``NeoHellenic'' fontundan al\i n\i r.
Bu font
\begin{align*}
\grm d&=\grm{\deltaalt},&
\grm E&=\grm{\Epsilonalt},&
\grm z&=\grm{\zetaalt},&
\grm X=\grm{\Xionealt}&=\grm{\Xitwoalt},&
\grm W&=\grm{\Omegaalt}
\end{align*}
alternatif bi\c cimlerini sa\u glar.
Sigman\i n k\"u\c c\"uk \gr c bi\c cimi
sadece bir s\"ozc\"u\u g\"un sonunda kullan\i l\i r.

\nocite{Pappus}
\nocite{MR654679}
\nocite{MR654680}
\nocite{MR0472307}
\nocite{Celgin}
\nocite{LSJ}
\nocite{Smyth}
\nocite{POCGD}
\nocite{Demirtas}
\nocite{Oklid-2014}
\nocite{MR17:814b}

\AfterBibliographyPreamble{\smaller}
%\bibliographystyle{plain}
%\bibliography{../../references}
%\bibliography{../references}

\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
  \def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{MR2093668}
Archimedes.
\newblock {\em The Two Books \emph{{O}n the Sphere and the Cylinder}}, volume~I
  of {\em The Works of {A}rchimedes}.
\newblock Cambridge University Press, Cambridge, 2004.
\newblock Translated into English, together with Eutocius' commentaries, with
  commentary, and critical edition of the diagrams, by Reviel Netz.

\bibitem{Ataturk}
Mustafa~Kemal Atat{\"u}rk.
\newblock {\em Geometri}.
\newblock T{\"u}rk Dil Kurumu, Ankara, 2000.
\newblock 4.\ bask{\i}. 1.\ bask{\i} 1971.

\bibitem{Celgin}
G{\"u}ler {\c C}elgin.
\newblock {\em Eski {Y}unanca--{T}{\"u}rk{\c c}e S{\"o}zl{\"u}k}.
\newblock Kabalc{\i}, {\.I}stanbul, 2011.

\bibitem{Demirtas}
Abdurrahman Demirta{\c s}.
\newblock {\em Matematik S{\"o}zl{\"u}{\u g}{\"u}}.
\newblock Bilim Teknik K{\"u}lt{\"u}r Yay{\i}nlar{\i}, Ankara, 1986.

\bibitem{MR17:814b}
Euclid.
\newblock {\em The Thirteen Books of {E}uclid's \emph{{E}lements}}.
\newblock Dover Publications, New York, 1956.
\newblock Translated from the text of {H}eiberg with introduction and
  commentary by Thomas L. Heath. In three volumes. Republication of the second
  edition of 1925. First edition 1908.

\bibitem{Oklid-2014}
Euclid.
\newblock {\em {\"O}klid'in \emph{{\"O}{\u g}eler}'inin 13 Kitab{\i}ndan
  Birinci Kitap}.
\newblock Mathematics Department, Mimar Sinan Fine Arts University, Istanbul,
  4th edition, September 2014.
\newblock The first of the 13 books of Euclid's Elements. Greek text, with
  Turkish version by {\"O}zer {\"O}zt{\"u}rk \&\ David Pierce.

\bibitem{MR654679}
Thomas Heath.
\newblock {\em A History of {G}reek Mathematics. {V}ol. {I}. {F}rom {T}hales to
  {E}uclid}.
\newblock Dover Publications Inc., New York, 1981.
\newblock Corrected reprint of the 1921 original.

\bibitem{MR654680}
Thomas Heath.
\newblock {\em A History of {G}reek Mathematics. {V}ol. {II}. From
  {A}ristarchus to {D}iophantus}.
\newblock Dover Publications Inc., New York, 1981.
\newblock Corrected reprint of the 1921 original.

\bibitem{MR0472307}
Morris Kline.
\newblock {\em Mathematical Thought from Ancient to Modern Times}.
\newblock Oxford University Press, New York, 1972.

\bibitem{LSJ}
Henry~George Liddell and Robert Scott.
\newblock {\em A {G}reek-{E}nglish Lexicon}.
\newblock Clarendon Press, Oxford, 1996.
\newblock Revised and augmented throughout by Sir Henry Stuart Jones, with the
  assistance of Roderick McKenzie and with the cooperation of many scholars.
  With a revised supplement.

\bibitem{POCGD}
James Morwood and John Taylor, editors.
\newblock {\em Pocket {O}xford Classical {G}reek Dictionary}.
\newblock Oxford University Press, Oxford, 2002.

\bibitem{Pappus}
{Pappus of Alexandria}.
\newblock {\em Pappus {A}lexandrini Collectionis Quae Supersunt}, volume~II.
\newblock Weidmann, Berlin, 1877.
\newblock E libris manu scriptis edidit, Latina interpretatione et commentariis
  instruxit Fridericus Hultsch.

\bibitem{Jones}
{Pappus of Alexandria}.
\newblock {\em Book 7 of the {\itshape Collection}. {P}art 1. {I}ntroduction,
  Text, and Translation}.
\newblock Springer Science+Business Media, New York, 1986.
\newblock Edited With Translation and Commentary by Alexander Jones.

\bibitem{MR1200456}
Proclus.
\newblock {\em A Commentary on the First Book of {E}uclid's \emph{{E}lements}}.
\newblock Princeton Paperbacks. Princeton University Press, Princeton, NJ,
  1992.
\newblock Translated from the Greek and with an introduction and notes by Glenn
  R. Morrow. Reprint of the 1970 edition. With a foreword by Ian Mueller.

\bibitem{MR0106139}
David~Eugene Smith.
\newblock {\em A Source Book in Mathematics}.
\newblock 2 vols. Dover Publications Inc., New York, 1959.
\newblock Unabridged republication of the first edition, 1929, published by
  McGraw-Hill.

\bibitem{Smyth}
Herbert~Weir Smyth.
\newblock {\em Greek Grammar}.
\newblock Harvard University Press, Cambridge, Massachussets, 1980.
\newblock Revised by Gordon M. Messing, 1956. Eleventh Printing. Original
  edition, 1920.

\bibitem{MR858706}
D.~J. Struik, editor.
\newblock {\em A Source Book in Mathematics, 1200--1800}.
\newblock Princeton Paperbacks. Princeton University Press, Princeton, NJ,
  1986.
\newblock Reprint of the 1969 edition.

\bibitem{Pascal-Coniques}
Rene Taton.
\newblock L' {\guillemotleft} {E}ssay pour les {C}oniques {\guillemotright} de
  {P}ascal.
\newblock {\em Revue d'histoire de science et de leurs applications},
  8(1):1--18, 1955.
\newblock
  \url{http://www.persee.fr/web/revues/home/prescript/article/rhs_0048-7996_1955_num_8_1_3488}.

\bibitem{MR13:419a}
Ivor Thomas, editor.
\newblock {\em Selections Illustrating the History of {G}reek Mathematics.
  {V}ol. {I}. {F}rom {T}hales to {E}uclid}.
\newblock Number 335 in Loeb Classical Library. Harvard University Press,
  Cambridge, Mass., 1951.
\newblock With an English translation by the editor.

\bibitem{MR13:419b}
Ivor Thomas, editor.
\newblock {\em Selections Illustrating the History of {G}reek Mathematics.
  {V}ol. {II}. {F}rom {A}ristarchus to {P}appus}.
\newblock Number 362 in Loeb Classical Library. Harvard University Press,
  Cambridge, Mass, 1951.
\newblock With an English translation by the editor.

\bibitem{Weeks-Adkins}
Arthur~W. Weeks and Jackson~B. Adkins.
\newblock {\em A Course in Geometry: Plane and Solid}.
\newblock Ginn and Company, Lexington, Massachusetts, 1970.

\end{thebibliography}
\clearpage
\thispagestyle{empty} % to get rid of heading from references
%\afterpage{%\clearpage
\begin{sidewaysfigure}
\addcontentsline{toc}{chapter}{\emph{Analiz Hazinesi}'nin i\c cindekiler}
\centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{*5l}
yazar&eserinin ad\i&Yunancas\i&\.Ingilizcesi&\\\hline
\"Oklid     &\emph{Veriler}           &\gr{Ded'wmena}         &\emph{Data}&1\\
Apollonius  &\emph{Oran\i n Kesilmesi}&\gr{L'ogou >apotom'h}  &\emph{Cutting-off of a Ratio}&2\\
Apollonius  &\emph{Alan\i n Kesilmesi}&\gr{Qwr'iou >apotom'h} &\emph{Cutting-off of an Area}&2\\
Apollonius  &\emph{Belirli Kesit}     &\gr{Diwrism'enhs tom'h}&\emph{Determinate Section}&2\\
Apollonius  &\emph{Te\u getler}       &\gr{>Epafa'i}          &\emph{Tangencies}&2\\
\"Oklid     &\emph{Porizmalar}        &\gr{Por'ismata}        &\emph{Porisms}&3\\
Apollonius  &\emph{Y\"onelmeler}      &\gr{Ne'useis}          &\emph{Vergings}&2\\
Apollonius  &\emph{D\"uzlem Yerleri}  &\gr{T'opoi >epip'edoi} &\emph{Plane Loci}&2\\
Apollonius  &\emph{Koni Kesitleri}    &\gr{Kwnika'i}          &\emph{Conics}&8\\
Aristaeus   &\emph{Cisim Yerleri}     &\gr{T'opoi stereo'i}   &\emph{Solid Loci}&5\\
\"Oklid     &\emph{Y\"uz Yerleri}     &\gr{T'opoi t~wn pr`os >epifane'ia|} &\emph{Surface Loci}&2\\
Eratosthenes&\emph{Ortalar hakk\i nda}&\gr{Per`i mesot'htai}  &\emph{On Means}&2
\end{tabular}
\caption*%[\emph{Analiz Hazinesi}'nin i\c cindekiler]%
{\emph{Analiz Hazinesi}'nin i\c cindekiler
(son s\"utun, eserin kitap [cilt] say\i s\i n\i\ verir)}\label{fig:treasury}
\end{sidewaysfigure}
%\clearpage}


\end{document}