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\begin{document}
\title{Koni kesitleri}
\author{David Pierce}
\date{25 Aral\i k 2012\\
  Geni\c sletilmi\c s, g\"ozden ge\c cirilmi\c s\\
22 Kas\i m 2018}
\publishers{Pop\"uler Matematik\\
  Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\url{dpierce@msgsu.edu.tr}\\
\url{mat.msgsu.edu.tr/~dpierce/}}

\maketitle

\begin{comment}
  \addchap*{\"Ozet}

  Neden koni kesitlerine parabol, hiperbol, veya elips denedi\u gi anlat\i l\i r.


  \eng{We explain why conic sections are called parabolas,
    hyperbolas, or ellipses.}
\end{comment}

\tableofcontents

\listoffigures

\chapter{Parabol, hiperbol, ve elips}

Bir \textbf{koni kesiti,}
bir koni y\"uzeyi ile bir d\"uzlemin kesi\c simidir.
Milattan \"once 3.\ y\"uzy\i lda Pergeli Apollonius, \nocite{Apollonius-Heiberg,MR1660991}
koni kesitlerine
\textbf{parabol, hiperbol,} ve \textbf{elips}
(\gr{parabol'h, <uperbol'h,} ve \gr{>'elleiyis}) adlar\i n\i\ verdi.
Bu adlar\i\ anlamak, amac\i m\i zd\i r.

Bir parabol\"un noktalar\i,
bir \textbf{odaktan} (\eng{focus})
ve bir \textbf{do\u grultmandan} (\eng{directrix})
ayn\i\ uzakl\i ktad\i r.
\"Orne\u gin dik eskenlerde
e\u ger
\Sekilde{fig:ip}ki gibi
bir parabol\"un
\begin{compactitem}
  \item
    oda\u g\i\ $(c,0)$ ve
  \item
    do\u grultman\i n\i n denklemi $x+c=0$
\end{compactitem}
ise,
o zaman parabol\"un noktalar\i,
a\c sa\u g\i daki denklemleri sa\u glar:
\begin{gather*}
  x+c=\sqrt{(x-c)^2+y^2},\\
  (x+c)^2=(x-c)^2+y^2,
%  x^2+2cx+c^2=x^2-2cx+c^2+y^2,\\
\end{gather*}
ve sonunda
\begin{equation*}
  4cx=y^2.
\end{equation*}
\begin{figure}[h]
  \centering
  \psset{unit=15mm}
  \begin{pspicture}(-1.2,-0.5)(3,3.5)
    %\psgrid
  \psset{plotpoints=100}
\psline{->}(0,-0.5)(0,3.5) % Y axis
\psline{->}(-1.2,0)(3,0) % X axis
\parametricplot[linewidth=2pt]{-0.5}{3.5}{t dup dup mul 4 div exch}
\psdots(1,0) % focus
\uput[d](1,0){$c$}
\psline[linewidth=2pt](-1,-0.5)(-1,3.5) % directrix
\uput[dl](-1,0){$-c$}
\uput[ul](1,0){$E$}
\psset{linestyle=dotted}
\psline(2.5,-0.5)(2.5,3.5)
\uput[d](1.5,0){$x$}
\psline(1.5,0)(! 1.5 dup sqrt 2 mul)(! 2.5 1.5 sqrt 2 mul)
\psset{linestyle=dashed}
%\psline(1,0)(0.25,1)(-1,1)
\uput[ul](! 0 1.5 sqrt 2 mul){$y$}
\uput[ur](! -1 1.5 sqrt 2 mul){$D$} % foot of perpendicular to directrix
\uput[u](! 1.5 dup sqrt 2 mul){$F$} % point on parabola
\psline(! -1 1.5 sqrt 2 mul)(! 1.5 dup sqrt 2 mul)(1,0)
\uput[ur](! 2.5 1.5 sqrt 2 mul){$G$}
\end{pspicture}

\caption[Parabol]{Parabol ($DF=EF$, $EF+FG$ sabittir)}
\label{fig:ip}

\end{figure}
Di\u ger koni kesitlerinin her birinin iki oda\u g\i\ vard\i r.
\begin{compactitem}
\item 
Elipste odaklardan uzakl\i klar\i n toplam\i\ sabittir.
\item
Hiperbolde odaklardan uzakl\i klar\i n fark\i\ sabittir.
\end{compactitem}
\c Sekiller \ref{fig:elips}
\begin{figure}
  \centering
  \psset{unit=18mm}
  \begin{pspicture}(-2,-2.2)(2,2.2)
    %\psgrid
\psset{plotpoints=100}
\psplot[linewidth=2pt]{-1.4}{1.4}{1 x x mul 4 div sub 3 mul sqrt}
\psplot[linewidth=2pt]{-1.4}{1.4}{1 x x mul 4 div sub 3 mul sqrt neg}
\parametricplot[linewidth=2pt]{-1.3}{1.3}{1 t t mul 3 div sub 4 mul sqrt t}
\parametricplot[linewidth=2pt]{-1.3}{1.3}{1 t t mul 3 div sub 4 mul sqrt neg t}
%\psline(-1,0)(0.5,1.68)(1,0)
\psline[linestyle=dashed](-1,0)(! 4 3 div 5 3 div sqrt)(1,0)
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\psline{->}(0,-2.2)(0,2.2)
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\uput[dr](2,0){$a$}
\uput[ul](! 0 3 sqrt){$b$}
\uput[ul](! 0 3 sqrt neg){$-b$}
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\uput[d](-1,0){$-c$}
\uput[ul](1,0){$E$}
\uput[u](-1,0){$D$}
\uput[ur](! 4 3 div 5 3 div sqrt){$F$}
\psset{linestyle=dotted}
\psline(! 4 3 div 0)(! 4 3 div 5 3 div sqrt)(! 0 5 3 div sqrt)
\uput[d](! 4 3 div 0){$x$}
\uput[l](! 0 5 3 div sqrt){$y$}
  \end{pspicture}
  \caption[Elips]{Elips ($DF+EF=2a$, $DE=2c$, $a^2=b^2+c^2$)}\label{fig:elips}
\end{figure}
ve \numarada{fig:hiperbol}ki gibi
\begin{figure}
  \centering
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  \begin{pspicture}(-3,-0.5)(4,3)
    %\psgrid
    \parametricplot[linewidth=2pt]{-0.5}3{1 t t mul 3 div add sqrt 2 mul t}
    \parametricplot[linewidth=2pt]{-0.5}2{1 t t mul 3 div add sqrt 2 mul neg t}
    \psline{->}(-3,0)(4,0)
    \psline{->}(0,-0.5)(0,3)
    \psdots(! 7 sqrt 0)(! 7 sqrt neg 0)
    \psline[linestyle=dashed](! 7 sqrt 0)(! 12 sqrt 6 sqrt)(! 7 sqrt neg 0)
    \uput[dr](! 12 sqrt 6 sqrt){$F$}
\uput[dr](-2,0){$-a$}
\uput[dl](2,0){$a$}
\uput[l](! 0 3 sqrt){$b$}
\uput[d](! 7 sqrt 0){$c$}
\uput[d](! 7 sqrt neg 0){$-c$}
\uput[ul](! 7 sqrt 0){$E$}
\uput[u](! 7 sqrt neg 0){$D$}
\psset{linestyle=dotted}
\psline(! 12 sqrt 0)(! 12 sqrt 6 sqrt)(! 0 6 sqrt)
\uput[d](! 12 sqrt 0){$x$}
\uput[l](! 0 6 sqrt){$y$}
%\psset{linestyle=dashed}
\psline(2,0)(! 2 3 sqrt)(! 0 3 sqrt)
\psline(0,0)(! 12 sqrt 3)
  \end{pspicture}

\caption[Hiperbol]{Hiperbol ($DF-EF=2a$, $DE=2c$, $a^2+b^2=c^2$)}
\label{fig:hiperbol}

\end{figure}
yukar\i daki
\begin{compactitem}
  \item
    toplam veya fark $2a$, ve
    \item
      odaklar $(-c,0)$ ve $(c,0)$,
\end{compactitem}
olsun.
E\u grinin bir denklemi
\begin{equation*}
  2a\mp\sqrt{(x-c)^2+y^2}=\sqrt{(x+c)^2+y^2},
\end{equation*}
ve kareler al\i nd\i\u g\i nda
\begin{gather*}
  4a^2\mp4a\sqrt{(x-c)^2+y^2}=4cx,\\
  a^2-cx=\pm a\sqrt{(x-c)^2+y^2},\\
  a^4-2a^2cx+c^2x^2=a^2x^2-2a^2cx+a^2c^2+a^2y^2,\\
  a^2(a^2-c^2)=(a^2-c^2)x^2+a^2y^2.
\end{gather*}
E\u ger
\begin{equation*}
  a^2-c^2=\pm b^2
\end{equation*}
ise, o zaman
\begin{equation}\label{eqn:standard}
  \frac{x^2}{a^2}\pm\frac{y^2}{b^2}=1.
\end{equation}

G\"ord\"um\"uz \"ozellikleri uyarak bir ip ile
koni kesitlerinin her birini \c cizebiliriz;
ama bu \"ozellikler, kesitlerin adlar\i n\i\ anlatmaz.

\chapter{Kelimeler}

Parabol, hiperbol, ve elips adlar\i n\i n normal anlamlar\i\ vard\i r.
%Bu anlamlar, yukar\i daki \"ozellikleri anlatmaz:
\begin{description}
  \item[\gr{b'allw}] (\emph{f{}iil}) atmak, f\i rlatmak.

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

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

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

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

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

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

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

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

\item[\gr{>'elleiyis}] (\emph{isim}) eksiklik.
\end{description}
Son kelime, \c Celgin'in \cite{Celgin} s\"ozl\"u\u g\"unde de\u gildir. 
%(Tam tan\i mlar i\c cin arkadaki eke bak\i n\i z.)

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

Koni kesitlerini anlatmak i\c cin
Apollonius
\begin{compactitem}
  \item
    denklemleri de\u gil,
  \item
    \emph{orant\i lar\i} kullan\i r.
\end{compactitem}
Bir orant\i,
\begin{compactitem}
\item
  iki \emph{oran\i n} bir ayn\i l\i\u g\i d\i r,
  \item
d\"ort \textbf{b\"uy\"ukl\"u\u g\"un} bir ba\u g\i nt\i s\i d\i r.
\end{compactitem}
S\i n\i rl\i\ do\u grular, alanlar, ve cisimler, b\"uy\"ukl\"uk \"ornekleridir.
Bir b\"uy\"ukl\"uk \textbf{\c co\u galt\i labilir.}
B\"oylece her $n$ sayma say\i s\i\ i\c cin,
bir $A$ b\"uy\"ukl\"u\u g\"un\"un
\begin{equation*}
nA
\end{equation*}
$n$ kat\i\ vard\i r.
$B$, bir b\"uy\"ukl\"uk daha olsun.
E\u ger bir $n$ say\i s\i\ i\c cin
\begin{align*}
nA&>B,&nB&>A
\end{align*}
ise,
o zaman b\"uy\"ukl\"uklerin \textbf{oran\i\ vard\i r.}
(Bu tan\i m, \"Oklid'in \emph{\"O\u geler}'inin
\textsc v.\ kitab\i n\i n 4.\ tan\i m\i d\i r.)
\nocite{Euclid-Heiberg,MR17:814b,bones,MR1932864}
$A$'n\i n $B$'ye oran\i\ i\c cin
\begin{align*}
&A/B,&A:B
\end{align*}
ifadeleri yaz\i labilir.

$A$ ve $B$'nin oran\i\ olsun;
$C$ ve $D$'nin oran\i\ olsun.
T\"um $n$ ve $m$ sayma say\i lar\i,
a\c sa\u g\i daki denklikleri sa\u glas\i n.
\begin{gather*}
	nA>mB\iff nC>mD,\\
	nA=mB\iff nC=mD,\\
	nA<mB\iff nC<mD.
\end{gather*}
O zaman
\begin{compactitem}
  \item
$A$'n\i n $B$'ye \textbf{oran\i,}
$C$'nin $D$'ye \textbf{oran\i\ ile ayn\i d\i r,} ve
\item
  $A$, $B$, $C$, ve $D$ b\"uy\"ukl\"ukleri,
  \textbf{orant\i l\i d\i r}
\end{compactitem}
    [\emph{\"O\u geler} \textsc v, 5.\ tan\i m].
    Bu durumda
\begin{align*}
	A:B&::C:D,&
	\frac AB&=\frac CD
\end{align*}
ifadeleri yaz\i labilir.
\emph{\"O\u geler}'de bir orant\i,
iki oran\i n \emph{e\c sitli\u gi} de\u gil,
\emph{ayn\i l\i\u g\i d\i r.}
Bir oran,
s\i ral\i\ bir b\"uy\"ukl\"uk ikilisinin
bir denklik s\i n\i f\i d\i r.
(Benzer bir \c sekilde pozitif bir kesir,
s\i ral\i\ bir sayma say\i s\i\ ikilisinin
bir denklik s\i n\i f\i d\i r.)

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

\begin{theorem}[\emph{\"O\u geler} \textsc{vi}.1]\label{thm:vi.1}
  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.
\end{theorem}
  \begin{figure}[h]
    \centering
  \begin{pspicture}(-0.5,-0.5)(3,2.5)
\pspolygon(0,0)(3,0)(2,2)
\psline(1,0)(2,2)
\uput[u](2,2){$A$}
\uput[d](0,0){$B$}
\uput[d](1,0){$C$}
\uput[d](3,0){$D$}
\end{pspicture}
\caption{Y\"uksekli\u ginin ayn\i\ olan \"u\c cgenler}\label{fig:vi.1}
  \end{figure}

\begin{proof}
  Orant\i n\i n tan\i m\i ndan
  \Sekilde{fig:vi.1}
\begin{equation*}
ABC:ACD::BC:CD.\qedhere
\end{equation*}
\end{proof}

\begin{theorem}[Tales Theoremi, \emph{\"O\u geler} \textsc{vi}.2]\label{thm:Tales}
  Bir \"u\c cgende
  bir do\u gru iki kenar\i\ orant\i l\i\ bir \c sekilde keser,
  ancak ve ancak bu do\u gru, tabana paraleldir.
\end{theorem}

\begin{figure}[h]
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\hfill
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\pspolygon(0,0)(3,0)(2,3)
\psline(2.5,1.5)(0.67,1)
\psline(0.67,1)(2.67,1)
\uput[u](2,3){$A$}
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[l](0.67,1){$D$}
\uput[r](2.67,1){$E$}
\uput[r](2.5,1.5){$F$}
\end{pspicture}
%\hfill\mbox{}
\caption{Tales Teoremi}\label{fig:tales}
\end{figure}
\begin{proof}
\Sekilde{fig:tales}
e\u ger $DE\parallel BC$ ise, o zaman
\begin{align*}
AD:DB
&::ADE:DBE&&\text{[\Teorem{thm:vi.1}]}\\
&::ADE:DCE&&\text{[\emph{\"O\u geler} \textsc i.37]}\\
&::AE:EC.&&\text{[\Teorem{thm:vi.1}]}
\end{align*}
Ters de do\u grudur,
\c c\"unk\"u $DF\nparallel BC$ ise,
o zaman $AF:FC$ ve $AE:EC$ oranlar\i\ birbirinden farkl\i d\i r.
\end{proof}

E\u ger iki \c cokgenin
\begin{compactitem}
  \item
    a\c c\i lar\i\ (her biri birine) e\c sit ve
    \item
      kenarlar\i\ orant\i l\i
\end{compactitem}
ise,
o zaman \c cokgenler
birbirine \textbf{benzerdir} [\emph{\"O\u geler} \textsc{vi}, 1.\ tan\i m].
\"Orne\u gin
\begin{compactitem}
  \item
\Sekilde{fig:tales}
%$DE\parallel BC$ durumunda,
$ABC$ ve $ADE$ \"u\c cgenleri
birbirine benzerdir.
\item
  \Sekilde{fig:rect}
\begin{figure}
  \centering
  %\psset{unit=0.67cm}
  \hfill
  \begin{pspicture}(-0.2,-0.5)(4,3.5)
    %\psgrid
\pspolygon(0,0)(3,0)(4,3)(1,3)
\uput[u](1,3){$A$}
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[u](4,3){$D$}
\end{pspicture}
\hfill
\begin{pspicture}(0,-0.5)(2.67,3.5)
\pspolygon(0,0)(2,0)(2.67,2)(0.67,2)
\uput[u](0.67,2){$E$}
\uput[d](0,0){$F$}
\uput[d](2,0){$G$}
\uput[u](2.67,2){$H$}
\end{pspicture}
\hfill\mbox{}
  \caption{Benzer paralelkenar}\label{fig:rect}
  
\end{figure}
e\u ger
\begin{align*}
  ABC&=EFG,&
    AB:BC&::EF:FG
\end{align*}
ise,
o zaman
$ABCD$ ve $EFGH$ paralelkenarlar\i\ da
birbirine benzerdir.
\end{compactitem}

\begin{theorem} [\emph{\"O\u geler} \textsc{vi}.24, 26]\label{thm:diag}
  Ayn\i\ a\c c\i da olan paralelkenarlar benzerdir ve benzer oturur,
  ancak ve ancak k\"o\c segenleri,
  ayn\i\ do\u grudad\i r.
\end{theorem}

\begin{figure}[h]
  \centering
  \psset{unit=8mm}
  \begin{pspicture}(-1,0)(8,3)
    %\psgrid
  \psset{PointSymbol=none,labelsep=2pt} \pstGeonode[PosAngle={180,-90,90},PointName={default,none,none}]
  (0,0)A(8,0)D(-1,3)E
  \pstTranslation[PosAngle=0]ADE[C]
  \ncline AD\ncline DC\ncline CE\ncline EA
  \pstHomO[HomCoef=0.6,PointName=none]A{D,E}
  \pstHomO[HomCoef=0.6,PosAngle=90]AC[B]
  \ncline{D'}B\ncline B{E'}
  \psset{linestyle=dashed}
  \ncline AC
\end{pspicture}

  \caption{Ayn\i\ a\c c\i da olan paralelkenarlar}\label{fig:diag}
  
\end{figure}
\"Orne\u gin \Sekilde{fig:diag},
$AB$ ile $AC$ paralelkenarlar\i\ birbirine benzerdir ve benzer oturur,
ancak ve ancak $A$, $B$, ve $C$ noktalar\i\ bir do\u grudad\i r.

\chapter{\c Cemberler}

\begin{theorem}[\emph{\"O\u geler} \textsc{iii}.1]\label{thm:iii.1}
  Bir \c cemberin bir kiri\c sinin orta dikmesi,
  \c cemberin merkezinden ge\c cer.
\end{theorem}

\begin{theorem}[\emph{\"O\u geler} \textsc{iii}.35]
  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.
  %  dolay\i s\i yla par\c calar orant\i l\i d\i r.
\end{theorem}

\begin{proof}
Diyoruz ki \Sekilde{fig:chords}
\begin{figure}
\centering  
\psset{unit=16mm}
\begin{pspicture}(-1,-2)(1,1)
  %\psgrid
 \SpecialCoor
 \pscircle(0,0){1}
 % \psline(1;-60)(1;120)
 \psline(1;-45)(1;135)
 \psline(1;-30)(1;210)
% \psline(1;210)(0,0)
% \psline(0,0)(0,-0.5)
 \begin{comment}
   \uput[120](0.5;210){$r$}
   \uput[d](! 3 sqrt 4 div neg -0.5){$b$}
   \uput[d](0.25,-0.5){$c$}
   \uput[dl](0.5;135){$r$}
   \uput[dl](0.35;-45){$a$}
   \uput[l](0,-0.25){$d$}
   \uput[u](0.68,-0.5){$b-c$}
   \uput[dl](0.85;-45){$r-a$}
 \end{comment}
\uput[210](1;210){$A$}
\uput[-30](1;-30){$B$}
\uput[135](1;135){$C$}
\uput[-45](1;-45){$D$}
\uput[247.5](0.5,-0.5){$E$}
%\psdots[dotsize=6pt](1;210)(1;-30)(1;135)(1;-45)(0.5,-0.5)
\end{pspicture}
\hfill
\psset{unit=32mm}
\begin{pspicture}(-1,-1)(1,1)
  %\psgrid
 \SpecialCoor
 \pscircle(0,0){1}
 % \psline(1;-60)(1;120)
 \psline(1;-45)(1;135)
 \psline(1;-30)(1;210)
 \psline(1;210)(0,0)
 \psline(0,0)(0,-0.5)
\uput[120](0.5;210){$r$}
\uput[d](! 3 sqrt 4 div neg -0.5){$b$}
\uput[d](0.25,-0.5){$c$}
\uput[dl](0.5;135){$r$}
\uput[dl](0.35;-45){$a$}
\uput[l](0,-0.25){$d$}
\uput[u](0.68,-0.5){$b-c$}
\uput[dl](0.85;-45){$r-a$}
\begin{comment}
  \uput[210](1;210){$A$}
  \uput[-30](1;-30){$B$}
  \uput[135](1;135){$C$}
  \uput[-45](1;-45){$D$}
  \uput[247.5](0.5,-0.5){$E$}
\end{comment}
%\psdots[dotsize=6pt](1;210)(1;-30)(1;135)(1;-45)(0.5,-0.5)
\end{pspicture}

  \caption{\c Cemberin iki kesi\c sen kiri\c sleri}\label{fig:chords}
  
\end{figure}
\begin{equation}\label{eqn:chords}
AE\cdot EB=CE\cdot ED.
\end{equation}
\begin{comment}
  \begin{gather}\label{eqn:chords}
    AE\cdot EB=CE\cdot ED,\\\label{eqn:chords-2}
    AE:CE::ED:EB.
  \end{gather}
\end{comment}
Zira kiri\c slerin birinin \c cemberin merkezinden
ge\c cti\u gini varsayabiliriz.
Bu kiri\c s $CD$ olsun.
\Sekilde{fig:chords}
Pisagor Teoremi [\emph{\"O\u geler} \textsc i.47] ile
\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*}
yani istedi\u gimiz \eqref{eqn:chords} denklemi sa\u glan\i r.
%Bu durumda \eqref{eqn:chords-2} orant\i s\i,
%\emph{\"O\u geler} \textsc{vi}.16 \"onermesi ile \c c\i kar.
\end{proof}

\"Ozel bir durum vard\i r.

\begin{theorem}\label{thm:semi}
  Yar\i m dairenin \c cap\i na bir dikmede olan kare,
  \c cap\i n par\c calar\i\ taraf\i ndan i\c cerilen dikd\"ortgene e\c sittir.
\end{theorem}

\"Orne\u gin \Sekilde{fig:radius}
\begin{figure}
  \centering
  \psset{unit=3cm}
  \begin{pspicture}(0,0.9)(2,2)
%    \psgrid
\psarc(1,1){1}0{180}
\pspolygon(0,1)(2,1)(0.4,1.8)
\psline(0.4,1.8)(0.4,1)
\uput[l](0,1){$A$}
\uput[r](2,1){$B$}
\uput[u](0.4,1.8){$C$}
\uput[d](0.4,1){$D$}
\end{pspicture}

  \caption{Yar\i\c cember}\label{fig:radius}
  
\end{figure}
\begin{equation*}
  CD^2=AD\cdot DB.
\end{equation*}
\begin{comment}
  \begin{align*}
    CD^2&=AD\cdot DB,&
    AD:CD&::CD:DB.
  \end{align*}
\end{comment}
(Sonu\c c olarak $ACB$ a\c c\i s\i\ dik olmal\i.)

\chapter{Uygulamalar}

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

\begin{problem}[\emph{\"O\u geler} \textsc i.44]\mbox{}
  
\centering
    \gr{Par`a t`hn doje~isan e>uje~ian}\\
    \gr{t~w| doj'enti  trig'wnw| >'ison}\\
    \gr{parallhl'ogrammon \underline{parabale~in}}\\
    \gr{>en  t~h| doje'ish| gwn'ia| e>ujugr'am\-mw|.}

    Verilen bir do\u gru boyunca\\
    verilen bir \"u\c cgene e\c sit\\
    bir paralelkenar \underline{uygulamak}\\%yerle\c stirmek\\
    verilen bir a\c c\i da. 
\end{problem}

\begin{problem}[\emph{\"O\u geler} \textsc i.45]\mbox{}

  \centering
    \gr{T~w| doj'enti e>ujugr'ammw| >'ison}\\
    \gr{parallhl'ogrammon sust\-'hsasjai}\\
    \gr{>en t~h| doje'ish| gwn'ia| e>ujugr'ammw|.}

    Verilen bir \c cokgene e\c sit\\
    bir paralelkenar in\c sa etmek\\
    verilen bir a\c c\i da. 
\end{problem}
%(Burada \emph{d\"uzkenar} bir \c sekil, bir \c cokgendir.) 
Sonu\c c olarak a\c sa\u g\i daki problemi \c c\"ozebiliriz:

\begin{problem}\label{prob}\mbox{}
  
\centering
Verilen bir do\u gru boyunca\\
verilen bir \c cokgene e\c sit\\
bir paralelkenar \underline{uygulamak}\\
verilen bir a\c c\i da. 
\end{problem}

\emph{\"O\u geler}'in \textsc{vi}.\ kitab\i nda \"Oklid,
Problem \numaranin{prob} son sat\i r\i nda iki de\u gi\c sikli\u gi yapar:

\begin{problem}[\emph{\"O\u geler} \textsc{vi}.28]\label{prob:leipw}\mbox{}
  
\centering

    \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 \c cokgene e\c sit\\
    bir paralelkenar \underline{uygulamak}\\%yerle\c stirmek\\
    verilen bir paralelkenara benzer kadar
    \underline{eksik kalan\vphantom p}.
\end{problem}

\"Orne\u gin
\begin{compactitem}
  \item
    verilen do\u gru $AB$,
  \item
    verilen \c cokgen $C$, ve
  \item
    verilen paralelkenar $D$
\end{compactitem}
ise, o zaman \Sekilde{fig:eksik}ki gibi
taban\i\ $AB$ olan bir $AE$ paralelkenar\i n\i n
\begin{compactitem}
\item
  $AH$ par\c cas\i\ $C$ \c cokgenine e\c sittir;
\item
  $GE$ par\c cas\i\ $D$ paralelkenar\i na benzerdir.
\end{compactitem}
\begin{figure}[h]
  \centering
  \begin{pspicture}(-1,-0.5)(6,2.5)
    \pspolygon(0,0)(6,0)(5,2)(-1,2)
    \psline(4,0)(3,2)
    \uput[d](0,0){$A$}
    \uput[d](6,0){$B$}
    \uput[u](5,2){$E$}
    \uput[u](-1,2){$F$}
    \uput[d](4,0){$G$}
    \uput[u](3,2){$H$}
  \end{pspicture}
  \caption{Eksikli\u gi olan bir paralelkenar\i n uygulamas\i}\label{fig:eksik}
  
\end{figure}
\"Oklid'in verdi\u gi bir \c sart\i\ yazmad\i m:
$C$, taban\i\ $AB$'nin yar\i s\i\ olan
ve $D$'ye benzer olan paralelkenardan daha b\"uy\"uk olamaz.
\begin{comment}
  (Problem \numarada{prob:leipw}n bir \c sart\i\ \c c\i kard\i m:
  Verilen\c cokgenin [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]
      .}})
\end{comment}

\begin{problem}[\emph{\"O\u geler} \textsc{vi}.29]\label{prob:hyper}\mbox{}

  \centering
    \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 \c cokgene e\c sit\\
    bir paralelkenar \underline{uygulamak}\\% yerle\c stirmek\\
    verilen paralelkenara benzer kadar \underline{a\c san}.
\end{problem}

\begin{figure}[h]
    \centering
    \psset{xunit=15mm}
    %\pstilt{80}{
    \begin{pspicture}(0,-0.5)(5,3.5)
      \psline(2,0)(2,3)(5,3)(5,0)(0,0)(0,1)(5,1)
      \psline(4,0)(4,3)
      \psline(2,3)(5,0)
      \uput[u](0,1){$A$}
      \uput[ur](4,1){$B$}
      \uput[ul](2,1){$E$}
      \uput[u](4,3){$F$}
      \uput[d](2,0){$H$}
      \uput[u](5,3){$K$}
      \uput[d](5,0){$L$}
    %  \uput[d](4,0){$M$}
    \end{pspicture}
    \caption{Uygulanan ve a\c san bir paralelkenar}\label{fig:hyper}
    
  \end{figure}
\begin{proof}[\"Oklid'in \c c\"oz\"um\"u.]\mbox{}
  \begin{compactdesc}
\item[A\c c\i klama.]
  Verilen do\u gru $AB$,
  \c cokgen $C$, ve
  paralelkenar $D$ olsun.
\item[Belirtme.]
\Sekilde{fig:hyper}ki gibi
\"oyle bir $AL$ paralelkenar\i n\i\ in\c sa edece\u giz ki
\begin{align*}
  AL&=C,& BL&\sim D.
\end{align*}
\item[D\"uzeltme.]\mbox{}
\begin{enumerate}
\item
  $AE=EB$ olsun
  [\emph{\"O\u geler} \textsc i.10].
\item
  $EF\sim D$ olsun
  [\emph{\"O\u geler} \textsc{vi}.18].
\item
  $HK=EF+C$ ve $HK\sim D$ olsun
%  $HK$ paralelkenar\i,
%  $EF$ paralelkenar\i\ ile $C$ \c seklinin toplam\i na e\c sit
%  ve $D$ paralelkenar\i na benzer olsun
  [\emph{\"O\u geler} \textsc{vi}.25].
\end{enumerate}
\item[G\"osterme.]\mbox{}
  \begin{enumerate}
\item
  $HB=BK$
  %paralelkenar\i na e\c sittir
  [\emph{\"O\u geler} \textsc i.43].
\item
  $AH=HB$.
\item
$AH=BK$.
\item
  $AL=HLKB$ \emph{gnomonu}
  (\gr{gn'wmwn}, %bilen;
  g\"une\c s saatinin g\"ostergesidir).
\item
  $HK=EF+HLKB$.
\item
$C=HLKB$.
\item
$AL=C$.
\item
  $BL\sim D$
  [\Teorem{thm:diag}].\qedhere
  \end{enumerate}
%\"Oyleyse $AL$ paralelkenar\i, istedi\u gimiz paralelkenard\i r.
  \end{compactdesc}
\end{proof}

\begin{figure}[h]
\centering
 \psset{xunit=15mm}
\begin{pspicture}(0,-0.33)(5,1)
\pspolygon(0,0)(5,0)(5,1)(0,1)
\psline(2,0)(2,1)
\psline(4,0)(4,1)
\uput[u](0,1){$A$}
\uput[u](4,1){$B$}
\uput[u](2,1){$E$}
\uput[u](1,0){$a$}
\uput[u](3,0){$a$}
\uput[u](4.5,0){$z$}
\uput[r](5,0.5){$cz/b$}
\end{pspicture}

  \caption{Kartezyen \c c\"oz\"um}\label{fig:Descartes}
  
\end{figure}
Problem \numaranin{prob:hyper}
Kartezyen \c c\"oz\"um\"u de vard\i r.
Verilen
\begin{compactitem}
\item
do\u grunun uzunlu\u gu $2a$,
\item
\c cokgenin alan\i\ $x^2$,
\item
paralelkenar\i n
\begin{compactitem}
\item
 eni $b$,
 \item
 y\"uksekli\u gi $c$
\end{compactitem}
\end{compactitem}
olsun.
  \.Istedi\u gimiz fazla gelen paralelkenar\i n eni, $z$ olsun
(\Sekle{fig:Descartes} bak\i n).
O zaman istedi\u gimiz paralelkenar\i n y\"uksekli\u gi, $cz/b$ olur,
ve
\begin{gather*}
  	x^2=(2a+z)\cdot\frac{cz}b,\\
	bx^2=cz(2a+z).
\end{gather*}
\"Oyleyse $z$ belirlenir.
Asl\i nda
%son denklemde
$z+a=y$ ise,
o zaman
\begin{align*}
z&=y-a,&
2a+z&=y+a,  
\end{align*}
dolay\i s\i yla
\begin{gather*}
bx^2=c(y^2-a^2),\\
cy^2-bx^2=a^2c,
\end{gather*}
ve sonunda
\eqref{eqn:standard} gibi
\begin{equation*}
  \frac{y^2}{a^2}-\frac{x^2}{a^2c/b}=1
\end{equation*}
hiperbol denklemi \c c\i kar.

\chapter{Koni kesitleri}

Bir \textbf{koni,}
\emph{tepesi} ve \emph{taban\i} taraf\i ndan belirlenir.
Koninin
\begin{compactitem}
  \item
    \textbf{taban\i,} bir dairedir;
  \item
    \textbf{tepesi,} taban\i n d\"uzleminde olmayan bir noktad\i r.
\end{compactitem}
O zaman koninin
\begin{compactitem}
  \item
\textbf{y\"uzeyi,}
tepeden taban\i n \c cevresine giden
t\"um do\u grular\i n noktalar\i ndan olu\c sur;
\item
\textbf{ekseni,}
taban\i n merkezinden tepeye giden do\u grudur.
\end{compactitem}
Eksen, tabana dik olmayabilir.
\"Orne\u gin \Sekilde{fig:cone}
\begin{compactitem}
\item
  tepesi $A$,
\item
  taban\i, \c cap\i\ $BC$ olan daire,
\item
  ekseni $FA$
\end{compactitem}
olan koni vard\i r.
$ABC$ \"u\c cgeni, koninin bir \textbf{eksen \"u\c cgenidir.}
\begin{figure}
  \psset{unit=22mm}
%  \subfloat[]{\label{fig:ax}
\begin{pspicture}(-1.1,-0.5)(1.1,2.3)
  \psset{Alpha=120}
\pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
\pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{90}A{$A$}
\pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
\ncline AB
\pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
\ncline AC\ncline BC
\pstThreeDPut(0,0,0){\pnode O}
\pstThreeDPut(1,0,0){\pnode X}
\psset{PointName=none,PointSymbol=none,linestyle=dotted}
\multido{\iv=0+30}{12}{
\pstHomO[HomCoef=\iv\space cos]OX[X_\iv]
\pstHomO[HomCoef=\iv\space sin]OC[Y_\iv]
\pstTranslation O{X_\iv}{Y_\iv}[Z_\iv]
\ncline A{Z_\iv}
}
\end{pspicture}
%  }
  \hfill
%  \subfloat[]{\label{fig:3}
  \begin{pspicture}(-1.1,-0.5)(1.1,2.3)
%    \psgrid
    \psset{Alpha=120,PointSymbol=none}
    \pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
    \pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{90}A{$A$}
    \pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
      \ncline AB
    \pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
      \ncline AC\ncline BC
    \pstMiddleAB BCF
    \ncline FA
  \end{pspicture}
  %}

  \caption{Koni, tepesi, taban\i, y\"uzeyi, ekseni, ve bir eksen \"u\c cgeni}
  \label{fig:cone}
  
\end{figure}
\begin{comment}
  \begin{figure}[h]
    \centering
    %  \psset{unit=0.75cm}
    \begin{pspicture}(-0.5,0)(4.5,4.5)
      \pstGeonode[PosAngle={90,180,0}](2.8,4)A(0,1)B(4,1)C
      \psellipse(2,1)(2,1)
      \pspolygon(0,1)(4,1)(2.8,4)
      %\uput[u](2.8,4){$A$}
      %\uput[l](0,1){$B$}
      %\uput[r](4,1){$C$}
      \psline(2,0)(2.8,4)(2,2)
      \psline(3,0.13)(2.8,4)(1,1.87)
      \psline(3.73,0.5)(2.8,4)(0.27,1.5)
      \psline(1,0.13)(2.8,4)(3,1.87)
      \psline(3.73,1.5)(2.8,4)(0.27,0.5)
    \end{pspicture}
    \hfill
    \begin{pspicture}(-0.5,0)(4.5,4.5)
      \psellipse(2,1)(2,1)
      \pspolygon(0,1)(4,1)(2.8,4)
      \uput[u](2.8,4){$A$}
      \uput[l](0,1){$B$}
      \uput[r](4,1){$C$}
      %\psline(2,0)(2.8,4)(2,2)
      %\psline(3,1.87)(3,0.13)
      %\uput[u](3,1.87){$D$}
      %\uput[dr](3,0.13){$E$}
      %\uput[ur](3,1){$G$}
      \psline(2,1)(2.8,4)
      \uput[d](2,1){$F$}
    \end{pspicture}

    \caption{Koni, tepesi, taban\i, y\"uzeyi, ekseni, ve bir eksen \"u\c cgeni}
    \label{fig:cone}
    
  \end{figure}
\end{comment}
Koninin taban\i n\i n bir $DE$ kiri\c si se\c cilmi\c s olsun.
Bu kiri\c sin orta dikmesinin $BC$ \c cap\i\ oldu\u gunu,
\Teoremi{thm:iii.1} kullanarak
varsayabiliriz.
Koninin taban\i n\i n $BC$ \c cap\i\ ve $DE$ kiri\c si,
\Sekilde{fig:base-chord}ki gibi
$G$ noktas\i nda kesi\c ssin. 
\begin{figure}
  \psset{unit=22mm}
%  \subfloat[]{\label{fig:ax}
\begin{pspicture}(-1.1,-0.5)(1.1,2.3)
  \psset{Alpha=120}
\pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
\pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{90}A{$A$}
\pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
\ncline AB
\pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
\ncline AC\ncline BC
\pstThreeDPut[SphericalCoor](1,150,0){\pnode D}\nput{90}D{$D$}
\pstThreeDPut[SphericalCoor](1,30,0){\pnode E}\nput{-45}E{$E$}
\ncline DE
\pstInterLL[PosAngle=-90,PointSymbol=none]BCDEG
\end{pspicture}
%  }
  \hfill
%  \subfloat[]{\label{fig:3}
  \begin{pspicture}(-1.1,-0.5)(1.1,2.3)
%    \psgrid
    \psset{Alpha=120,PointSymbol=none}
%  \pstThreeDCoor[xMax=1.2,yMax=1.2,zMax=1.2]
    \pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
    \pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{90}A{$A$}
    \pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
      \ncline AB
    \pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
      \ncline AC\ncline BC
    \pstThreeDPut[SphericalCoor](1,150,0){\pnode D}
      \nput{45}D{$D$}
    \pstThreeDPut[SphericalCoor](1,30,0){\pnode E}
        \nput{-45}E{$E$}\ncline DE
    \pstInterLL[PosAngle=-90]BCDEG
    \pstTranslation[PointName=none]CAG
    \pstInterLL[PosAngle=105]ABG{G'}H
    \ncline GH
    \pstTranslation[PointName=none] GDH
%   \psset{PointSymbol=*}
    % begin multido
    \newcommand{\points}{9}
    \multido{\i=1+1}{\points}{
      \pstHomO[PointName=none,
        HomCoef=2 \i\space mul \points\space 1 add div 1 sub]
          H{H'}[Y_\i]
      \pstHomO[PointName=none,
        HomCoef=\pstDistAB H{Y_\i}\space \pstDistAB H{H'}\space div dup mul]
              HG[X_\i]
      \pstTranslation[PointName=none,
                  PosAngle=45]H{X_\i}{Y_\i}[Z_\i]
      } % end multido
   \pstGenericCurve[GenCurvFirst=E,GenCurvLast=D]{Z_}1{\points}
  \end{pspicture}
  %}

    \caption{Koninin taban\i n\i n bir kiri\c si;
      parabol ve diyametresi}\label{fig:base-chord}
  
\end{figure}

Bir d\"uzlem,
$DE$ kiri\c sinden ge\c csin.
Bu d\"uzlem,
eksen \"u\c cgenini bir $GH$ do\u grusunda keser.
$H$ noktas\i n\i n $AB$ do\u grusunda oldu\u gunu
varsayabiliriz. 
D\"uzlem, koninin y\"uzeyini bir $DHE$ e\u grisinde keser.
Bu e\u gri, bir \textbf{koni kesitidir,}
ve $HG$,
koni kesitinin bir \textbf{diyametresidir} (\gr{di'ametron}),
\c c\"unk\"u $DE$ kiri\c sine paralel olan kiri\c sleri ikiye b\"oler.
\begin{comment}
  \Sekle{fig:diyametre} bak\i n.
  \begin{figure}[h]
    \centering
    %  \psset{plotpoints=100}
    \begin{pspicture}(0,-1.75)(5.0625,2.25)
      \psplot{0.25}{5.0625}{x sqrt}
      \psplot{0.25}{3.0625}{x sqrt neg}
      \parametricplot{-0.5}{0.5}{t dup dup mul exch}
      \psline(! 1 16 div 1 4 div)(! 1 16 div 4 add 1 4 div)
      \psline(! 0.25 1 sqrt sub dup dup mul exch)
      (! 0.25 1 sqrt add dup dup mul exch)
      \psline(! 0.25 2 sqrt sub dup dup mul exch)
      (! 0.25 2 sqrt add dup dup mul exch)
      \psline(! 0.25 3 sqrt sub dup dup mul exch)
      (! 0.25 3 sqrt add dup dup mul exch)
      \psline(! 0.25 4 sqrt sub dup dup mul exch)
      (! 0.25 4 sqrt add dup dup mul exch)
      \uput[r](! 0.25 4 sqrt add dup dup mul exch){$D$}
      \uput[r](! 0.25 4 sqrt sub dup dup mul exch){$E$}
      \uput[l](! 1 16 div 1 4 div){$H$}
      \uput[r](! 1 16 div 4 add 1 4 div){$G$}
    \end{pspicture}
    \caption{Koni kesiti ve bir diyametre}\label{fig:diyametre}
    
  \end{figure}

\end{comment}
\.Iki durum vard\i r.
\begin{compactenum}
\item
  E\u ger $GH$ diyametresi $AC$ do\u grusuna paralel ise,
  o zaman kesit bir \textbf{parabold\"ur.}
Tekrar \Sekle{fig:base-chord} bak\i n.
\item
  Di\u ger durumda, uzat\i l\i nca  $GH$,
  $AC$ do\u grusunu bir $K$ noktas\i nda keser.
  O zaman $HK$ do\u grusunun $Z$ orta noktas\i,
  koni kesitinin \textbf{merkezidir.}
Bu merkez,
\begin{compactitem}
\item
  koninin i\c cinde ise,
  kesit bir \textbf{elipstir;}
\item
  koninin d\i\c s\i nda ise,
  kesit bir \textbf{hiperbold\"ur.} 
\end{compactitem}
\Sekle{fig:central} bak\i n.
\begin{figure}[h]
  \psset{unit=22mm}
      \begin{pspicture}(-1.1,-1.3)(1,1.5)
%    \psgrid
    \psset{Alpha=-36.5,Beta=33,%PointSymbol=none
    }
    \pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
    \pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{90}A{$A$}
    \pstThreeDPut(0,-1,0){\pnode B}\nput{45}B{$B$}
    \ncline AB
    \pstThreeDPut(0, 1,0){\pnode C}\nput{225}C{$C$}
    \ncline BC
    \pstThreeDPut[SphericalCoor](1,130,0){\pnode D}\nput{-60}D{$D$}
    \pstThreeDPut[SphericalCoor](1,50,0){\pnode E}\nput{-180}E{$E$}
    \ncline DE
    \psset{PointSymbol=none}
    \pstInterLL[PosAngle=10]BCDEG
    \pstHomO[PosAngle=70,HomCoef=.35]AB[H]
    \pstInterLL[PosAngle=-110]GHACK
    \ncline KH\ncline AK
    \pstMiddleAB[PosAngle=20,PointSymbol=*] HKZ
    \psset{PointName=none}
    \pstTranslation[PosAngle=180,PointName=none] GDZ
      \pstHomO[PosAngle=-30,
      HomCoef=\pstDistAB ZH\space dup
      dup mul \pstDistAB ZG\space dup mul sub sqrt div]Z{Z'}[M]
      \pstHomO[PosAngle=-75,HomCoef=-1]ZM[N]
%      \ncline MN
      \newcommand{\points}{15}
     % begin multido
      \multido{\i=1+1}{\points}{
        \pstHomO[PointName=none,
          PosAngle=225,
          HomCoef=\i\space \points\space 1 add div 360 mul cos]
        ZM[Y_\i]
        \pstHomO[PointName=none,
          PosAngle=135,%PointSymbol=default,
          HomCoef=\i\space \points\space 1 add div 360 mul sin]
                ZH[X_\i]
        \pstTranslation[PointName=none,
          PosAngle=45]Z{X_\i}{Y_\i}[Z_\i]
      } % end multido
      \pstGenericCurve[GenCurvFirst=M,GenCurvLast=M]{Z_}1{\points}
      \end{pspicture}
      \hfill
        \begin{pspicture}(-1,-0.6)(1,2.6)
    %\psgrid
    \psset{Alpha=120,PointSymbol=none
    }
    \pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
    \pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{0}A{$A$}
    \pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
    \ncline AB
    \pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
    \ncline BC
   \pstThreeDPut[SphericalCoor](1,210,0){\pnode D}\nput{135}D{$D$}
   \pstThreeDPut[SphericalCoor](1,-30,0){\pnode E}\nput{-90}E{$E$}
    \ncline DE
    \pstInterLL[PosAngle=-90]BCDEG
    \pstHomO[PosAngle=120,HomCoef=.35]AB[H]
    \pstInterLL[PosAngle=80]HGACK
    \ncline GK\ncline CK
    \pstMiddleAB[PosAngle=165,PointSymbol=*] HKZ
    \psset{PointName=none}
    \pstTranslation[PosAngle=180,PointName=none] GDZ
      \pstHomO[PosAngle=135,
      HomCoef=\pstDistAB ZH\space dup
     dup mul \pstDistAB ZG\space dup mul exch sub sqrt div]Z{Z'}[M]
      \pstHomO[PosAngle=-45,HomCoef=-1]ZM[N]
%      \ncline MN
      % begin multido
      \newcommand{\points}{19}
      \multido{\i=1+1}{\points}{
        \pstHomO[PointName=none,PosAngle=225,
          HomCoef=2 \i\space mul \points\space 1 add div 1 sub]
            Z{Z'}[Y_\i]
        \pstHomO[PointName=none,PosAngle=135,
          HomCoef=\pstDistAB Z{Y_\i}\space
          \pstDistAB ZM div dup mul 1 add sqrt]
                ZH[X_\i]
        \pstTranslation[PointName=none,PosAngle=45]
          Z{X_\i}{Y_\i}[Z_\i]
      } % end multido
      \pstGenericCurve[GenCurvFirst=E,GenCurvLast=D]{Z_}1{\points}
  \end{pspicture}

  \caption{Merkezli koni kesitleri ve diyametreleri}\label{fig:central}
  
\end{figure}
\end{compactenum}

Koni kesitimiz, hiperbol olsun.
\Teoreme{thm:semi} g\"ore
\begin{equation*}
DG^2=BG\cdot GC.
\end{equation*}
Tabana paralel olan bir d\"uzlem, koniyi kessin.
Bu d\"uzlem, $HG$ do\u grusunu bir $L$ noktas\i nda keser,
ve eksen \"u\c cgenini bir $MN$ do\u grusunda keser.
Ayr\i ca d\"uzlem, koninin y\"uzeyini bir \c cemberde keser.
\c Sekiller \ref{fig:hyp-in-cone} ve \numaraya{fig:plan} bak\i n.
O zaman koni kesitinin d\"uzlemi,
bu \c cemberi bir $PQ$ kiri\c sinde keser.
$MN$ do\u grusu, \c cemberin bir \c cap\i d\i r
ve $PQ$ kiri\c sinin orta dikmesidir.
\"Ozel olarak,
\begin{equation*}
  PL=LQ.
\end{equation*}
%$HG$ do\u grusu, koni kesitinin bir diyametresidir.
Teorem \ref{thm:semi} sayesinde
\begin{equation*}
\frac{DG^2}{PL^2}
=\frac{BG\cdot GC}{ML\cdot LN}
=\frac{BG}{ML}\cdot\frac{GC}{LN},
\end{equation*}
ve sonu\c c olarak, \Teoremde{thm:Tales}n
\begin{equation}\label{eqn:squares}
\frac{DG^2}{PL^2}
=\frac{HG}{HL}\cdot\frac{KG}{KL}.
\end{equation}
\begin{figure}
  \centering
  \psset{unit=40mm}
          \begin{pspicture}(-1,-0.6)(1,2.6)
    %\psgrid
    \psset{Alpha=120,PointSymbol=none
    }
    \pstThreeDCircle(0,0,0)(1,0,0)(0,1,0)
    \pstThreeDPut(-0.3,0.8,2){\pnode A}\nput{0}A{$A$}
    \pstThreeDPut(0,-1,0){\pnode B}\nput{210}B{$B$}
    \ncline AB
    \pstThreeDPut(0, 1,0){\pnode C}\nput{ 30}C{$C$}
    \ncline BC
   \pstThreeDPut[SphericalCoor](1,210,0){\pnode D}\nput{135}D{$D$}
   \pstThreeDPut[SphericalCoor](1,-30,0){\pnode E}\nput{-90}E{$E$}
    \ncline DE
    \pstInterLL[PosAngle=-90]BCDEG
    \pstHomO[PosAngle=120,HomCoef=.35]AB[H]
    \pstInterLL[PosAngle=80]HGACK
    \ncline GK\ncline CK
   \psset{PointName=none}
    \pstMiddleAB[PosAngle=165] HKZ
    \pstTranslation[PosAngle=180,PointName=none] GDZ
      \pstHomO[PosAngle=135,
      HomCoef=\pstDistAB ZH\space dup
     dup mul \pstDistAB ZG\space dup mul exch sub sqrt div]Z{Z'}[M]
      \pstHomO[PosAngle=-45,HomCoef=-1]ZM[N]
%      \ncline MN
      % begin multido
      \newcommand{\points}{19}
      \multido{\i=1+1}{\points}{
        \pstHomO[PosAngle=225,
          HomCoef=2 \i\space mul \points\space 1 add div 1 sub]
            Z{Z'}[Y_\i]
        \pstHomO[PosAngle=135,
          HomCoef=\pstDistAB Z{Y_\i}\space
          \pstDistAB ZM div dup mul 1 add sqrt]
                ZH[X_\i]
        \pstTranslation[PosAngle=45]
          Z{X_\i}{Y_\i}[Z_\i]
      } % end multido
      \pstGenericCurve[GenCurvFirst=E,GenCurvLast=D]{Z_}1{\points}
      \ncline{Z_5}{Z_15}
      \pstTranslation BC{X_5}
      \psset{PointName=default}
      \pstInterLL[PosAngle=200] AB{X_5}{X_5'}M
      \pstInterLL AC{X_5}{X_5'}N
      \ncline MN
      \nput{45}{X_5}{$L$}
      \nput{45}{Z_5}{$Q$}
      \nput{135}{Z_15}{$P$}
   \psset{PointName=none}
   \pstMiddleAB MNO
   \pstTranslation GDO
   \pstThreeDPut(1,0,0){\pnode P}
   \pstInterLL APO{O'}Q
   \multido{\i=0+1}{13}{
   \pstHomO[HomCoef=30 \i\space mul cos]OQ[X_\i]
   \pstHomO[HomCoef=30 \i\space mul sin]OM[Y_\i]
   \pstTranslation O{X_\i}{Y_\i}[Z_\i]}
   \pstGenericCurve{Z_}0{12}
  \end{pspicture}

  \caption{Konide hiperbol}\label{fig:hyp-in-cone}
  
\end{figure}
\begin{figure}[h]
  \begin{pspicture}(-0.2,-2)(3,4.5)
    %\psgrid
\psline(1,0)(3,4)(3,0)(0,0)(3,3)
%\psline(1,1)(3,1)
%\psline(1,0)(0.2,0.4)(3,4)
%\psline(1.5,1)(0.9,1.3)
%\psline(2,2)(1.6,2.2)
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[r](3,3){$A$}
\uput[ul](2,2){$H$}
\uput[d](1,0){$G$}
\uput[u](3,4){$K$}
\psline(1,1)(3,1)
\uput[dr](1.5,1){$L$}
\uput[ul](1,1){$M$}
\uput[ul](3,1){$N$}
\end{pspicture}
\hfill
\begin{pspicture}(-0.5,-2)(2,2)
\pscircle(1,1){1}
\psline(0,1)(2,1)
\psline(0.5,0.13)(0.5,1.87)
\uput[u](0.5,1.87){$P$}
\uput[ur](0.5,1){$L$}
\uput[d](0.5,0.13){$Q$}
\uput[l](0,1){$M$}
\uput[r](2,1){$N$}
\end{pspicture}
\hfill
\begin{pspicture}(0,-0.5)(3.5,3)
%  \psgrid
\pscircle(1.5,1.5){1.5}
\psline(0,1.5)(3,1.5)
\psline(1,0.09)(1,2.91)
\uput[u](1,2.91){$D$}
\uput[ur](1,1.5){$G$}
\uput[d](1,0.09){$E$}
\uput[l](0,1.5){$B$}
\uput[r](3,1.5){$C$}
\end{pspicture}
  \caption{Koninin kesen d\"uzlemleri}\label{fig:plan}
  
\end{figure}

\c Simdi
\Sekilde{fig:upright}ki gibi
Problem \numarayi{prob} kullanarak
\begin{align*}
  GR\cdot GH&=DG^2,&
LS\cdot LH&=PL^2
\end{align*}
olsun.
O zaman \eqref{eqn:squares} e\c sitli\u ginden
\begin{equation*}
\frac{RG}{SL}=\frac{KG}{KL},
\end{equation*}
dolay\i s\i yla (\Teoremde{thm:diag}n)
$R$, $S$, ve $K$, bir do\u grudad\i r.
\begin{figure}[h]
  \centering
  \psset{unit=15mm}
\begin{pspicture}(-0.3,-0.3)(3.3,4.3)
\psline(1,0)(3,4)(3,0)(0,0)(3,3)
\psline(1,1)(3,1)
\psline(1,0)(0.2,0.4)%(3,4)
\psline(1.5,1)(0.9,1.3)
\psline(2,2)(1.6,2.2)
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[r](3,3){$A$}
\uput[dr](2,2){$H$}
\uput[d](1,0){$G$}
\uput[u](3,4){$K$}
\psline(1,1)(3,1)
\uput[dr](1.5,1){$L$}
\uput[d](1,1){$M$}
\uput[r](3,1){$N$}
\uput[l](0.2,0.4){$R$}
\uput{1pt}[l](0.9,1.3){$S$}
%\uput[u](1.6,2.2){$T$}
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\psline(0.9,1.3)(1.4,2.3)
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\end{pspicture}
\hfill
\begin{pspicture}(-0.3,-0.3)(3.3,4.3)
\psline(1,0)(3,4)(3,0)(0,0)(3,3)
%\psline(1,1)(3,1)
\psline(1,0)(0.2,0.4)(3,4)
%\psline(1.5,1)(0.9,1.3)
\psline(2,2)(1.6,2.2)
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[r](3,3){$A$}
\uput[-75](2,2){$H$}
\uput[d](1,0){$G$}
\uput[u](3,4){$K$}
%\uput[dr](1.5,1){$L$}
%\uput[d](1,1){$M$}
%\uput[r](3,1){$N$}
\uput[l](0.2,0.4){$R$}
%\uput{1pt}[l](0.9,1.3){$S$}
\uput[u](1.6,2.2){$T$}
\psline(1.6,2.2)(1.2,2.4)(0.2,0.4)
%\psline(0.9,1.3)(1.4,2.3)
\psline(1.6,2.2)(2.6,4.2)(3,4)
\uput[r](3,2){$U$}
\uput[d](2.5,2){$V$}
\psline(3,2)(2,2)
\psline(3,3)(2.5,2)
\end{pspicture}

  \caption{Hiperbol\"un dik kenar\i}\label{fig:upright}
  
\end{figure}

\c Simdi $KG$ do\u grusuna $H$ noktas\i ndaki dikme, $RK$ do\u grusunu $T$ noktas\i nda kessin.  O zaman Problem \numarada{prob:hyper}ki gibi
\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}
Bu y\"uzden $DHE$ koni kesitimize \textbf{hiperbol} (\gr{<uperbol'h}) denir.  
Bu tan\i m,
Pergeli Apollonius'\"un \emph{Koni kesitleri} \cite{Apollonius-Heiberg,MR1660991} eserinin
\textsc i.\ kitab\i n\i n 12.\ \"onermesinde bulunur.
Bu eserin \textsc{iii}.\ kitab\i nda odak \"ozelli\u gi anlat\i l\i r.
$HT$ do\u grusuna \textbf{dik kenar} (%
\gr{>orj'ia pleur'a}%
,
\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*}
O zaman \Sekilde{fig:upright}ki gibi
$AV\parallel KH$ ise
\begin{align*}
  \frac{BG}{GH}&=\frac{HV}{VA},&
  \frac{GC}{GK}&=\frac{VU}{VA},
\end{align*}
ve sonu\c c olarak
\begin{equation*}
\frac{HT}{HK}=\frac{HV\cdot VU}{VA^2}.
\end{equation*}

Hiperbol\"un $GE$ ve $LQ$ gibi yar\i kiri\c sleri,
\textbf{ordinatlard\i r,}
ve onlara paralel olan ve $H$ noktas\i ndan ge\c cen do\u gru,
hiperbole te\u gettir.
%\c Simdi $GE$,
%hiperbol\"un rasgele bir ordinat\i\ olarak d\"u\c s\"un\"ulebilir.
%Bu $GE$ ordinat\i\ taraf\i ndan
Hiperbol\"un diyametresinden kesilmi\c s $HG$ par\c cas\i,
$GE$ ordinat\i na uyan
\textbf{apsistir}
(\eng{abscissa,} bir \c seyden kesilmi\c s  bir \c sey;
%Apollonius'un kelimesi
\gr{>apolambanom'enh},
 bir \c seyden al\i nmi\c s  bir \c sey \cite{MR3312989}).
 $HK$ do\u grusu, hiperbol\"un \textbf{\c capraz kenar\i d\i r}
 (\gr{plag'ia pleur'a}, \eng{latus transversum}).

 \c Simdi hiperbol\"un $HT$ dik kenar\i n\i\
 ve $HK$ \c capraz kenar\i n\i\ kullanarak
 $GE$ ordinat\i n\i n ve $GH$ apsisinin ili\c skilerini
 a\c sa\u g\i daki gibi ifade edebiliriz.
 E\u ger
 \begin{equation*}
   GR:HT::GK:HK
 \end{equation*}
 ise, o zaman
 \begin{equation*}
   	GE^2=GR\cdot GH.
 \end{equation*}
\Sekle{fig:curve} bak\i n.
\begin{figure}[h]
  \centering
  \psset{unit=1.4cm,
    %unit=6mm    ,
    plotpoints=100,labelsep=5pt}
\begin{pspicture}(-5.2,-6.29)(2,3.96)
  %\psgrid
  \psset{linestyle=dotted}
\psline(-2,-3.46)(0.83,-3.46)
\psline(2,3.46)(1.13,3.96)(-2.87,-2.96)
\psline(-3.73,-2.46)(2,3.46)
\psset{linestyle=solid}
\pspolygon(-2,-3.46)(-2,-6.29)(-4.83,-6.29)(-4.83,-3.46)
\psline(-2,-3.46)(-3.73,-2.46)(-1.73,1)(-0.87,0.5)
\uput[r](2,3.46){$K$}
\uput[u](-0.87,0.5){$T$}
\uput[ul](-3.73,-2.46){$R$}
\uput[u](0,0){$H$}
\uput[l](-4.83,-3.46){$E$}
\uput[dr](0.83,-3.46){$D$}
\uput[dr](-2,-3.46){$G$}
\psset{linewidth=2pt}
\psline(-2,-3.46)(2,3.46)
\psline(0,0)(-0.87,0.5)
\psset{linewidth=3pt}
\parametricplot04{t 4 div 4 t add mul sqrt t 60 cos mul sub
0 t 60 sin mul sub}
\parametricplot04{0 t 4 div 4 t add mul sqrt sub t 60 cos mul sub
0 t 60 sin mul sub}
\end{pspicture}
\caption{Hiperbol}\label{fig:curve}
\end{figure}

\begin{comment}

\appendix

\chapter{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}

%\bibliographystyle{plain}
%\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}{1}

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\newblock Teubner, 1974.
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\bibitem{Celgin}
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\newblock {\em Eski {Y}unanca--{T}{\"u}rk{\c c}e S{\"o}zl{\"u}k}.
\newblock Kabalc{\i}, {\.I}stanbul, 2011.

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\bibitem{bones}
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\newblock Green Lion Press, Santa Fe, NM, 2002.
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\bibitem{MR1932864}
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\end{thebibliography}


\end{document}
