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\usepackage{amsmath}

\begin{document}
\title{Koni kesitleri}
\author{David Pierce}
\date{25 Aral\i k 2012}
\publishers{Pop\"uler Matematik, Matematik B\"ol\"um\"u, MSGS\"U\\
%Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle

\begin{abstract}
Neden koni kesitlerine parabol, hiperbol, veya elips denir.%edi\u gi anlat\i l\i r.

\eng{Why conic sections are called parabolas, hyperbolas, or ellipses.}% is explained.}
\end{abstract}

\tableofcontents

\section{Parabol, hiperbol, ve elips}

Bir \textbf{koni kesiti,} bir koni y\"uzeyi ile bir d\"uzlemin kesi\c
simi olan bir e\u gridir.  Milattan \"once 3.\ y\"uzy\i lda Pergeli
Apollonius \cite{Apollonius-Heiberg,MR1660991}, koni kesitlerine
\textbf{parabol, hiperbol,} ve \textbf{elips} (yani \gr{parabol'h,
  <uperbol'h,} ve \gr{>'elleiyis}) adlar\i n\i\ verdi.  Bu e\u grileri
ip kullanarak \c cizebiliriz: 

\begin{minipage}{0.3\textwidth}
Elipse iki \textbf{odaktan} giden uzakl\i klar\i n toplam\i\ de\u gi\c smez.
\end{minipage}
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$\begin{gathered}
	a^2-b^2=c^2,\\
	\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.
\end{gathered}$

\begin{minipage}{0.3\textwidth}
Hiperbole iki odaktan giden uzakl\i klar\i n fark\i\ de\u gi\c smez.
\end{minipage}
\hfill
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$\begin{gathered}
	a^2+b^2=c^2,\\
	\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.
\end{gathered}$

\begin{minipage}{0.3\textwidth}
Parabole bir odaktan ve \textbf{do\u grultmandan} giden uzakl\i klar\i\ birbirine e\c sittir, dolay\i s\i yla odaktan ve do\u grultmana paralel olan bir do\u grudan e\u griye giden uzakl\i klar\i n toplam\i\ de\u gi\c smez.
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$\begin{gathered}
	2a-b=2c,\\
y=\frac{x^2}{4c}.
\end{gathered}$


\section{Kelimeler}

Parabol, hiperbol, ve elips adlar\i n\i n normal anlamlar\i\ vard\i r.  Bu anlamlar, yukar\i daki \"ozellikleri anlatmaz:

\gr{b'allw} (\emph{f{}iil}) atmak, f\i rlatmak.

\gr{bol'h} (\emph{isim}) atma, f\i rlatma.

\gr{parab'allw} (\emph{f{}iil}) kar\c s\i la\c st\i rmak, yakla\c smak.

\gr{parabol'h} (\emph{isim}) kar\c s\i la\c st\i rma, yakla\c sma, ili\c ski, benzeme, benzerlik.

\gr{<uperb'allw} (\emph{f{}iil}) ge\c cmek, a\c smak.

\gr{<uperbol'h} (\emph{isim}) ge\c cme, a\c sma, fazlal\i k.

\gr{le'ipw} (\emph{f{}iil}) b\i rakmak, eksik olmak.

\gr{le~iyis} (\emph{isim}) eksiklik.

\gr{>elle'ipw} (\emph{f{}iil}) bir yana b\i rakmak, arkada b\i rakmak.

\gr{>'elleiyis} (\emph{isim}) eksiklik (bu kelime, \cite{Celgin}
kayna\u g\i nda de\u gil). 

%(Tam tan\i mlar i\c cin arkadaki eke bak\i n\i z.)

\section{Oranlar, orant\i lar, ve benzerlik}

S\i n\i rl\i\ do\u grular, alanlar, ve cisimler, \textbf{b\"uy\"ukl\"uk} \"ornekleridir.

Bir b\"uy\"ukl\"uk \textbf{\c co\u galt\i labilir.}  Her $n$ tamsay\i s\i\ i\c cin, bir b\"uy\"ukl\"u\u g\"un $n$ kat\i\ vard\i r.  B\"uy\"ukl\"uk $A$ ise, onun $n$ kat\i,
\begin{equation*}
nA
\end{equation*}
olarak yaz\i labilir.

$A$ ile $B$, b\"uy\"ukl\"uk olsunlar.  E\u ger bir $n$ tamsay\i s\i\ i\c cin,
\begin{align*}
nA&>B,&nB&>A
\end{align*}
olursa,
o zaman b\"uy\"ukl\"uklerin \textbf{oran\i} vard\i r.  (Bu tan\i m, \"Oklid'in \emph{\"O\u geler}'inin V.\ kitab\i n\i n 4.\ tan\i m\i d\i r \cite{Euclid-Heiberg,MR17:814b,bones,MR1932864}.)  $A$ b\"uy\"ukl\"u\u g\"un\"un $B$ b\"uy\"ukl\"u\u g\"une oran\i\ i\c cin
\begin{align*}
&A/B,&A:B
\end{align*}
yaz\i labilir.  Ancak bir oran \c cizilemez: soyuttur.  (Asl\i nda bir denklik s\i n\i f\i\ olacak.)

$A$ ile $B$ b\"uy\"ukl\"uklerinin oran\i\ olsun, ve
$C$ ile $D$ b\"uy\"ukl\"uklerinin oran\i\ olsun.  T\"um $n$ ve $m$ tamsay\i lar\i\ i\c cin
\begin{gather*}
	nA>mB\iff nC>mD,\\
	nA=mB\iff nC=mD,\\
	nA<mB\iff nC<mD
\end{gather*}
oldu\u gunu varsayal\i m.
O zaman $A$ b\"uy\"ukl\"u\u g\"un\"un $B$ b\"uy\"ukl\"u\u g\"une oran\i,
$C$ b\"uy\"ukl\"u\u g\"un\"un $D$ b\"uy\"ukl\"u\u g\"une oran\i\ ile \textbf{ayn\i d\i r.}  %(Oranlar sadece e\c sit de\u gil; ayn\i d\i r.)
Ayr\i ca $A$, $B$, $C$, ve $D$ b\"uy\"ukl\"ukleri, \textbf{orant\i l\i d\i r} [\emph{\"O\u geler} V, 5.\ tan\i m], ve
bu durumda
\begin{align*}
	A:B&::C:D,&
	\frac AB&=\frac CD
\end{align*}
ifadeleri yaz\i labilir.  (\"Onceki ifadeyi tercih ederim, \c c\"unk\"u bir orant\i da oranlar sadece e\c sit de\u gil, ayn\i d\i r, ama iki b\"uy\"ukl\"u\u g\"un kendileri, ayn\i\ olmadan e\c sit olabilir.)  Orant\i, oranlar\i\ olan b\"uy\"ukl\"uk \c ciftleri aras\i nda bir denklik ba\u g\i nt\i s\i d\i r.

\emph{\"O\u geler}'in V.\ kitab\i nda kolay kurallar g\"osterilir, mesela
\begin{equation*}
A:B::C:D\implies A+B:B::C+D:D\And B:A::D:C.
\end{equation*}

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\end{minipage}
\hfill
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Y\"uksekli\u ginin ayn\i\ olan \"u\c cgenlerin oran\i, tabanlar\i n\i n oran\i yla ayn\i d\i r [\emph{\"O\u geler} VI.1]:
\begin{equation*}
ABC:ACD::BC:CD.
\end{equation*}
\end{minipage}

\begin{minipage}{0.65\textwidth}
Dolay\i s\i yla, bu \c sekilde, $DE\parallel BC$ ise, o zaman
\begin{align*}
AD:DB
&::ADE:DBE\\
&::ADE:DCE\quad\text{[I.37]}\\
&::AE:EC,
\end{align*}
ve tersi de do\u grudur  [\emph{\"O\u geler} VI.2].
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\end{pspicture}
\end{minipage}

Bu \c sekilde, $DE\parallel BC$ durumunda, $ABC$ ile $ADE$ \"u\c cgenleri, birbirine \textbf{benzerdir} [VI, 1.\ tan\i m].

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\begin{minipage}{0.45\textwidth}
$ABCD$ ile $EFGH$ paralelkenarlar\i nda 
$ABC$ ile $EFG$ a\c c\i lar\i\ birbirine e\c sit ise ve $AB:BC::EF:FG$ ise,
o zaman paralelkenarlar birbirine \textbf{benzerdir} [VI, 1.\ tan\i m].
\end{minipage}

\begin{minipage}{0.6\textwidth}
Bu \c sekilde $AB$ ile $AC$ paralelkenarlar\i\ birbirine benzerdir ancak ve ancak $A$, $B$, ve $C$ noktalar\i\ bir do\u grudad\i r [VI.24, 26].
\end{minipage}
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\section{\c Cemberler}

Bir \c cemberin bir kiri\c sinin orta dikmesi, \c cemberin merkezinden ge\c cer [\emph{\"O\u geler} III.1].

\begin{minipage}{0.7\textwidth}
Bir \c cemberin iki kiri\c si kesi\c sirse, birinin par\c calar\i\ taraf\i ndan i\c cerilen dikd\"ortgen, \"otekinin par\c calar\i\ taraf\i ndan i\c cerilen dikd\"ortgene e\c sittir [III.35], ve o zaman par\c calar orant\i l\i d\i r [VI.16]:
\begin{gather*}
AE\cdot EB=CE\cdot ED,\\
AE:CE::ED:EB.
\end{gather*}
\end{minipage}
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Bunu g\"ostermek i\c cin, bir kiri\c sin \c cemberin merkezinden ge\c cti\u gini varsayabiliriz ve Pisagor Teoremini [I.47] kullanabiliriz:
\begin{align*}
(r+a)(r-a)
&=r^2-a^2\\
&=b^2+d^2-(d^2+c^2)\\
&=b^2-c^2\\
&=(b+c)(b-c).
\end{align*}
\end{minipage}

\begin{minipage}{0.65\textwidth}
\"Ozel olarak yar\i m dairenin \c cap\i na dikmedeki kare, \c cap\i n par\c calar\i\ taraf\i ndan i\c cerilen dikd\"ortgene e\c sittir:
\begin{align*}
CD^2&=AD\cdot DB,&
AD:CD&::CD:DB.
\end{align*}
(Sonu\c c olarak $ACB$ a\c c\i s\i\ dik olmal\i.)
\end{minipage}
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\section{Uygulamalar}

\"Oklid'de \gr{parab'allw} f{}iili, \emph{yerle\c stirmek} veya \emph{uygulamak} olarak \c cevrilebilir.  (\.Ingilizcede \eng{apply} olarak \c cevrilir.)  Bu kelime, birka\c c tane \emph{problemlerde} kullan\i l\i r:

\begin{center}
\"Onerme I.44:

  \begin{tabular}{rl}
\gr{Par`a t`hn doje~isan e>uje~ian}&Verilen bir do\u gru boyunca\\
\gr{t~w| doj'enti  trig'wnw| >'ison}&verilen bir \"u\c cgene e\c sit\\
\gr{parallhl'ogrammon \underline{parabale~in}}&bir paralelkenar \underline{uygulamak}\\%yerle\c stirmek\\
\gr{>en  t~h| doje'ish| gwn'ia| e>ujugr'am\-mw|.}&verilen bir
d\"uzkenar a\c c\i da. 
  \end{tabular}

\"Onerme I.45:

\begin{tabular}{rl}
\gr{T~w| doj'enti e>ujugr'ammw| >'ison}&Verilen bir d\"uzkenar
   [\c sekl]e e\c sit\\
\gr{parallhl'ogrammon sust\-'hsasjai}&bir paralelkenar in\c sa etmek\\
\gr{>en t~h| doje'ish| gwn'ia| e>ujugr'ammw|.}&verilen bir
d\"uzkenar a\c c\i da. 
\end{tabular}
\end{center}
(Burada bir d\"uzkenar a\c c\i, \c cokgen olarak d\"u\c s\"unebilir.) 
Sonu\c c olarak a\c sa\u g\i daki problemi \c c\"ozebiliriz:
\begin{center}
Verilen bir do\u gru boyunca\\
verilen bir d\"uzkenar [\c sekl]e e\c sit\\
bir paralelkenar \underline{uygulamak}\\
verilen bir d\"uzkenar a\c c\i da. 
\end{center}

VI.\ kitab\i nda \"Oklid, son sat\i rda de\u gi\c siklikler yapar:
\begin{center}
\"Onerme VI.28:

\gr{Par`a t`hn doje~isan e>uje~ian\\
t~w| doj'enti e>ujugr'ammw| >'ison\\
parallhl'ogrammon \underline{parabale~in}\\
\underline{>elle~ipon\vphantom b} e>'idei parallhlogr'ammw| <omo'iw| t~w| doj'enti.}

Verilen bir do\u gru boyunca\\
verilen bir d\"uzkenar [\c sekl]e e\c sit\\
bir paralelkenar \underline{uygulamak}\\%yerle\c stirmek\\
verilen [paralelkenar]a benzer bir paralelkenar \c sekli kadar \underline{eksik
kalan\vphantom p}.
\end{center}

(Buradan bir \c sart \c c\i kard\i m:
Verilen d\"uzkenar [\c sekl]in [verilen do\u grunun] yar\i [s\i n]da \c cizilmi\c s eksi\u ge benzer [\c sekil]den b\"uy\"uk olmamas\i\ gerekir.\footnote{\gr{de~i d`e
t`o did'omenon e>uj'ugrammon
%[<~w| de~i >'ison parabale~in]
m`h me~izon e>~inai
to~u >ap`o t~hc <hmise'iac >anagrafom'enou
<omo'iou t~w|  >elle'immati
%[to~u te >ap`o t~hc <hmise'iac ka`i <~w| de~i <'omoion >elle'ipein]
.}})

\begin{center}
\"Onerme VI.29:

\gr{Par`a t`hn doje~isan e>uje~ian\\
t~w| doj'enti e>ujugr'ammw| >'ison\\
parallhl'ogrammon \underline{parabale~in}\\
\underline{<uperb'allon} e>'idei parallhlogr'ammw| <omo'iw| t~w| doj'enti.}

Verilen bir do\u gru boyunca\\
verilen bir d\"uzkenar [\c sekl]e e\c sit\\
bir paralelkenar \underline{uygulamak}\\% yerle\c stirmek\\
verilen [paralelkenar]a benzer bir paralelkenar \c sekli kadar \underline{a\c san}.
\end{center}

\begin{proof}[Son \"onermenin g\"osterisi.]
Verilen do\u gru, $AB$ olsun; verilen d\"uzkenar \c sekil, $C$ olsun; verilen paralelkenar, $D$ olsun.
\begin{compactenum}
\item
$E$ noktas\i, $AB$ do\u grusunu ikiye b\"ols\"un.\hfill[I.10]
\begin{equation*}
AE=EB.
\end{equation*}
\item
$EF$ paralelkenar\i, $D$ paralelkenar\i na benzer olsun.\hfill[VI.18]
\begin{equation*}
EF\sim D.
\end{equation*}
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\uput[dl](5,3){$K$}
\uput[ur](5,0){$L$}
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%}
\item
$HK$ paralelkenar\i, $EF$ paralelkenar\i\ ile $C$ \c seklinin toplam\i na e\c sit ve $D$ paralelkenar\i na benzer olsun.\hfill[VI.25]
\begin{align*}
HK&=EF+C,&HK&\sim D.
\end{align*}
\item
$HB$ paralelkenar\i, $BK$ paralelkenar\i na e\c sittir.\hfill[I.43]
\begin{equation*}
HB=BK.
\end{equation*}
\item
$AH$ paralelkenar\i, $BK$ paralelkenar\i na e\c sittir.
\begin{equation*}
AH=BK.
\end{equation*}
\item
$AL$ paralelkenar\i, $HLKB$ \emph{gnomonuna} e\c sittir (\gr{gn'wmwn} bilen; g\"une\c s saatinin g\"ostergesi).
\begin{equation*}
AL=HLKB.
\end{equation*}
\item
Bu gnomon, $C$ \c sekline e\c sittir.
\begin{align*}
EF+C&=HK=EF+HLKB,&C&=HLKB.
\end{align*}
\item
$AL$ paralelkenar\i, $C$ \c sekline e\c sittir.
\begin{equation*}
AL=C.
\end{equation*}
\item
$AL$ paralelkenar\i, $AB$ do\u grusunu $D$ paralelkenar\i na benzer $BL$ paralelkenar\i\ kadar a\c sar.\hfill[I.24]
\begin{equation*}
BL\sim D.\qedhere
\end{equation*}
\end{compactenum}
%\"Oyleyse $AL$ paralelkenar\i, istedi\u gimiz paralelkenard\i r.
\end{proof}

\begin{proof}[Kartezyen \c c\"oz\"um.]\mbox{}
\begin{compactitem}
\item
Verilen do\u grunun uzunlu\u gu, $a$ olsun.
\item
Verilen d\"uzkenar \c seklin alan\i, $x^2$ olsun.  
\item
Verilen paralelkenar\i n
\begin{compactitem}
\item
 eni, $b$ olsun; 
 \item
 y\"uksekli\u gi, $c$ olsun.
 \hfill
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\uput[d](4.5,0){$z$}
\uput[r](5,0.5){$\displaystyle\frac{cz}b$}
\end{pspicture}
\end{compactitem}
\item
  \.Istedi\u gimiz fazla gelen paralelkenar\i n eni, $z$ olsun. 
\end{compactitem}
 O zaman y\"uksekli\u gi, $cz/b$, ve
\begin{align*}
	x^2&=(2a+z)\cdot\frac{cz}b,&
	bx^2&=cz\cdot(2a+z).
\end{align*}
\"Oyleyse $z$ belirlenir.
\end{proof}

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\hfill
\begin{minipage}{0.5\textwidth}
Burada $y=z+a$ ise, o zaman
\begin{gather*}
z=y-a,\\
2a+z=y+a,\\
bx^2=cy^2-a^2c,\\
cy^2-bx^2=a^2c,\\
\frac{y^2}{a^2}-\frac{x^2}{a^2c/b}=1.
\end{gather*}
Son denklemler, bir \textbf{hiperbol\"un} denklemleridir.
\end{minipage}

\section{Koni kesitleri}

\psset{labelsep=3pt}
\begin{minipage}{0.6\textwidth}
$A$ bir nokta, ve $BC$, bir dairenin \c cap\i\ olsun.  $A$ noktas\i, dairenin
d\"uzleminde olmas\i n.  O zaman \"oyle bir \textbf{koni} vard\i r ki 
\begin{compactitem}
\item
$A$ noktas\i, koninin \textbf{tepesidir,} ve
\item
$BC$ dairesi, koninin \textbf{taban\i d\i r.}
\end{compactitem}
\end{minipage}
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Koninin \textbf{y\"uzeyi,} tepeden taban\i n \c cevresine giden t\"um do\u grular\i n noktalar\i ndan olu\c sur (veya o do\u grular\i\ i\c cerir, veya o do\u grular taraf\i ndan olu\c sturulur).

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\hfill
\begin{minipage}{0.6\textwidth}
Bir d\"uzlem, koninin taban\i n\i\ $DE$ kiri\c sinde kessin.
Bu kiri\c sin orta dikmesinin $BC$ \c cap\i\ oldu\u gunu
varsayabiliriz.\hfill [III.3]
\end{minipage}

\begin{minipage}{0.6\textwidth}
$F$ noktas\i, $BC$ \c cap\i n\i n orta noktas\i\ olsun.
O zaman $AF$ do\u grusu, koninin \textbf{eksenidir.}
$ABC$ \"u\c cgeni, koninin bir \textbf{eksen \"u\c cgenidir.}
Koninin taban\i n\i n $BC$ \c cap\i\ ve $DE$ kiri\c si, $G$ noktas\i
nda kesi\c ssin. 
\end{minipage}
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\begin{minipage}{0.6\textwidth}
Koninin taban\i n\i\ kesen d\"uzlem, eksen \"u\c cgenini $GH$ do\u
grusunda kessin; $H$ noktas\i n\i n $AB$ do\u grusunda oldu\u gunu
varsayabiliriz. 

Kesen d\"uzlem, koninin y\"uzeyini bir $DHE$ e\u grisinde keser.  Bu e\u
gri, bir \textbf{koni kesitidir.}

Bu kesit, $A$ noktas\i n\i\ i\c cerirse, o zaman $H$ ile $A$,
ayn\i\ noktad\i r, ve kesit, $AD$ ile $AE$ do\u
grular\i ndan olu\c sur.
\end{minipage}

\begin{compactitem}
\item
E\u ger $GH$ do\u grusu, $AC$ do\u grusuna paralel ise, kesit bir
    \textbf{parabold\"ur.}
\item
Kalan durumda, uzat\i l\i rsa, $GH$ ile $AC$ do\u grular\i, bir $K$
noktas\i nda kesi\c sir.
Bu nokta,
\begin{compactitem}
\item
 koninin taban\i n\i n alt\i ndaysa, kesit bir
\textbf{elipstir,}
\item
koninin tepesinin \"ust\"undeyse, kesit bir
\textbf{hiperbold\"ur.} 
\end{compactitem}
\end{compactitem}

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\begin{minipage}{0.45\textwidth}
Koni kesitimiz, hiperbol olsun.
$DG$ do\u grusundaki kare, $BG$ ile $GC$ taraf\i ndan i\c cerilen dikd\"ortgene e\c sittir [III.35], yani
\begin{equation*}
DG^2=BG\cdot GC.
\end{equation*}
\end{minipage}
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\c Simdi tabana paralel olan bir d\"uzlem, koniyi kessin.  $HG$ do\u grusunu $L$ noktas\i nda kessin, ve eksen \"u\c cgenini $MN$ do\u grusunda kessin.

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O zaman koninin y\"uzeyi, bir \c cemberde kesilir, ve koni kesitinin d\"uzlemi, bu \c cemberi bir $PQ$ kiri\c sinde keser.  $MN$ do\u grusu, bu \c cemberin bir \c cap\i d\i r ve $PQ$ kiri\c sinin orta dikmesidir.

\"Ozel olarak, $PL=LQ$ y\"uz\"unden, $HG$ do\u grusu, koni kesitinin bir \textbf{\c cap\i d\i r.}  A\c sa\u g\i daki e\c sitli\u gimiz vard\i r.
\begin{equation*}
\frac{DG^2}{PL^2}
=\frac{BG\cdot GC}{ML\cdot LN}
=\frac{BG}{ML}\cdot\frac{GC}{LN}
=\frac{HG}{HL}\cdot\frac{KG}{KL}.
\end{equation*}

\begin{minipage}{0.65\textwidth}
\c Simdi $GR$ do\u grusu, $KG$ do\u grusuna dik olsun, ve $RH$ dikd\"ortgeni, $DG$ do\u grusundaki kareye e\c sit olsun, yani
\begin{equation*}
RG\cdot GH=DG^2
\end{equation*}
olsun.
Benzer \c sekilde $SH$ dikd\"ortgeni, $PL$ do\u grusundaki kareye e\c sit olsun, yani
\begin{equation*}
SL\cdot LH=PL^2
\end{equation*}
olsun.  O zaman
\begin{equation*}
\frac{RG}{SL}=\frac{KG}{KL},
\end{equation*}
dolay\i s\i yla $R$, $G$, ve $K$, bir do\u grudad\i r.
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\c Simdi $KG$ do\u grusuna $H$ noktas\i ndaki dikme, $RK$ do\u grusunu $T$ noktas\i nda kessin.  O zaman
\begin{verse}
Verilen $HT$ do\u grusuna\\
$DG$ do\u grusundaki kareye e\c sit olan\\
$RH$ dikd\"ortgenini uyguland\i k\\
verilen $KT$ dikd\"ortgenine benzer bir dikd\"ortgen kadar a\c san (\gr{<uperb'allon}).\nocite{Apollonius-Heiberg}
\end{verse}
\end{minipage}

Bu y\"uzden $DHE$ koni kesitine \textbf{hiperbol} (\gr{<uperbol'h}) denir.  
Bu tan\i m ve ispat, Pergeli Apollonius'\"un \emph{Koni kesitleri} \cite{Apollonius-Heiberg,MR1660991} eserinin I.\ kitab\i n\i n 12.\ \"onermesinde bulunur.  O eserin III.\ kitab\i nda odak \"ozelli\u gi anlat\i l\i r.

$HT$ do\u grusuna, \textbf{dik kenar} (\gr{>orj'ia}, \eng{latus rectum}) denir.  A\c sa\u g\i daki gibi bulunur:
\begin{equation*}
\frac{HT}{HK}=\frac{RG}{GK}=\frac{RG\cdot GH}{GK\cdot GH}=\frac{BG\cdot GC}{GK\cdot GH} =\frac{BG}{GH}\cdot\frac{GC}{GK}.\phantom{{}=\frac{HT}{HK}}
\end{equation*}
\begin{minipage}{0.8\textwidth}
O zaman $AV\parallel KH$ ise
\begin{equation*}
\frac{HT}{HK}=\frac{HV}{VA}\cdot\frac{VU}{VA}=\frac{HV\cdot VU}{VA^2}.
\end{equation*}
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\begin{gather*}
	GR:HT::GK:HK,\\
	EG^2=GR\cdot GH.
\end{gather*}

\begin{comment}

\appendix

\section{Eski Yunanca kelimelerinin tam tan\i mlar\i}

A\c sa\u g\i daki tan\i mlar, G\"uler \c Celgin'in \emph{Eski Yunanca--T\"urk\c ce S\"ozl\"uk} kitab\i ndan al\i nm\i\c st\i r \cite{Celgin}.


\gr{b'allw} f\i rlatmak, atmak; b\i rakmak; vurmak, darbe indirmek, \c carpmak; kovmak, p\"usk\"urtmek; yakalamak; yaralamak $\parallel$ \emph{med.} -i kendisi i\c cin atmak, f\i rlatmak.

\gr{<h bol'h} f\i rlatma, atma; (bitkiler i\c cin) \c ci\c ce\u gini d\"okme.

\gr{parab'allw} \eng{(throw beside [as fodder to animals], set
  beside)} atmak;\\
   emanet etmek, b\i rakmak;\\
    yol g\"ostermek, k\i lavuzluk etmek, rehberlik etmek;\\
     tehlikeye atmak, tehlikeye sokmak, tehlikeye maruz b\i rakmak;\\
      aldatmak;\\
       k\i yaslamak [k\i yas \eng{comparison, analogy, syllogism, deductive reasoning,} k\i yaslamak \eng{compare}], kar\c s\i la\c st\i rmak, mukayese [\eng{comparison}] etmek;\\
        yakla\c smak;\\
         sunmak $\parallel$
\emph{med.}\ yakla\c smak, yak\i n\i na gitmek, i\c cine girmek;\\
paralel hale getirmek;\\
 g\"ot\"urmek, k\i lavuzluk etmek;\\
(\emph{intrans.}) kendini tehlikeye atmak. 

\gr{<h parabol'h} \eng{(throwing beyond, excess)} kar\c s\i la\c st\i
rma, k\i yaslama;\\
ili\c ski, benzeme, benzerlik;\\
\c carp\i\c sma, kar\c s\i la\c sma;\\
do\u gru yoldan ay\i rma 

\gr{<uperb'allw} -in \"ust\"une, \"otesine f\i rlatmak;\\
-i ge\c cmek;\\
amac\i n\i\ a\c smak [\eng{cross, traverse, overcome, exceed, overtake, copulate with}], \"ol\c c\"uy\"u a\c smak $\parallel$
\emph{med.}\ abartmak, m\"ubala\u ga etmek, oldu\u gundan b\"uy\"uk
g\"ostermek. 

\gr{<h <uperbol'h} -in \"ust\"une, \"otesine f\i rlatma;\\
-i a\c sma, ge\c cme;\\
\"ol\c c\"uy\"u a\c sma, a\c s\i r\i l\i k, fazlal\i k;\\
s\"ure m\"uhlet. 

\gr{le'ipw} b\i rakmak, terk etmek, arkas\i nda b\i rakmak;\\ 
ihanet etmek, y\"uz\"ust\"u b\i rakmak;\\
gitmek, uzakla\c smak;\\
yetersiz olmak, eksik olmak, noksan olmak;\\
durmak, dinmek $\parallel$ \emph{med.}\ arkas\i nda b\i rakmak, geride b\i rakmak, (\emph{gen.}\ ile) -den yoksun, mahrum kalmak.

\gr{<h le~iyis} eksiklik, noksanl\i k;\\
yokluk, olmay\i\c s, bulunmay\i\c s.

\gr{>elle'ipw} bir yana b\i rakmak, arkada b\i rakmak;\\
ihmal etmek, savsaklamak;\\
ald\i rmamak, ald\i r\i\c s etmemek, \"onemsememek, \"onem vermemek, bo\c s vermek. 

\gr{>'elleiyis} [yok]

\end{comment}

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%\bibliography{../references}

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\begin{thebibliography}{1}

\bibitem{Apollonius-Heiberg}
{Apollonius of Perga}, \emph{Apollonii {P}ergaei qvae {G}raece exstant cvm
  commentariis antiqvis}, vol.~I, Teubner, 1974, Edidit et Latine interpretatvs
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\bibitem{MR1660991}
\bysame, \emph{Conics. {B}ooks {I}--{III}}, revised ed., Green Lion Press,
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\bibitem{Celgin}
G{\"u}ler {\c C}elgin, \emph{Eski {Y}unanca--{T}{\"u}rk{\c c}e s{\"o}zl{\"u}k},
  Kabalc{\i}, {\.I}stanbul, 2011.

\bibitem{Euclid-Heiberg}
Euclid, \emph{Euclidis {E}lementa}, Euclidis Opera Omnia, vol.~I, Teubner,
  1883, Edidit et Latine interpretatvs est I. L. Heiberg.

\bibitem{MR17:814b}
\bysame, \emph{The thirteen books of {E}uclid's {E}lements translated from the
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\bibitem{bones}
\bysame, \emph{The bones}, Green Lion Press, Santa Fe, NM, 2002, A handy
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\bibitem{MR1932864}
\bysame, \emph{Euclid's {E}lements}, Green Lion Press, Santa Fe, NM, 2002, All
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\end{thebibliography}


\end{document}