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%textwidth 1576pt
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\usepackage{amsmath,bm}
%\renewcommand{\theequation}{\fnsymbol{equation}}

\begin{document}
\title{Hiperbol}
\author{David Pierce}
\date{\today}
%\maketitle

\begin{minipage}[t]{\Awidth}
\VERYHuge\gr{UPERBOLH}\Large

\c Sekillerde $AB$ e\u grisi,
\begin{compactitem}
  \item
\textcolor{\upright}{\textbf{dikey kenar\i} $BC$} olan,
\item
\textcolor{\transverse}{\textbf{yanlamas\i na kenar\i} $BD$} olan, ve
\item
\textbf{merkezi} $E$ olan 
\end{compactitem}
\textbf{hiperbold\"ur,}
ve
\begin{compactitem}
  \item
$AF$, hiperbol\"un bir \textbf{ordinat\i d\i r,}
\item
$BF$, ona kar\c s\i l\i k gelen \textbf{absistir.}
\end{compactitem}

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\rput(0.2071,-0.5){$\ell c$}
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$%\displaystyle
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\normalsize
Bu durumda
\begin{equation*}
  \begin{gathered}[b]
  BC=\ell,\\
EB=a,
  \end{gathered}
  \qquad
BF=c,\qquad
FA=d\qquad\phantom{FA=ddddddd}
\end{equation*}
ise
\begin{equation}\label{eqn:Apollo}
\text{\Large\colorbox{\areas}{$d^2=\ell c+\frac{\ell}{2a}\cdot c^2$.}}
\end{equation}
\"Ozel olarak $d^2$ karesi, $\ell c$ dikd\"ortgenini a\c sar.
Bu nedenle Pergeli Apollonius, e\u griye
\gr{<uperbol'h}
(``a\c sma'') der.
\eqref{eqn:Apollo} e\c sitli\u ginden
\begin{equation}\label{eqn:center-d}
  d^2%=\ell c+\frac{\ell}{2a}\cdot c^2
%=\frac{\ell}{2a}\cdot (2ac+c^2)
=\frac{\ell}{2a}\cdot\bigl((a+c)^2-a^2\bigr).
\end{equation}
\eqref{eqn:Apollo} ve \eqref{eqn:center-d} e\c sitlikleri
tamamen geneldir.  
T\"um ordinatlar birbirine paraleldir.
%E\u ger
$GH$ herhangi bir ordinat ve
\begin{equation*}
\text{\colorbox{\xystcolor}{\color{white}$\bm{BH=x,\qquad HG=y}$}}
\end{equation*}
ise \eqref{eqn:center-d} e\c sitli\u gine g\"ore
\begin{equation*}
%\text{\large{$
y^2=\frac{\ell}{2a}\bigl((a+x)^2-a^2\bigr),%$}}
\end{equation*}
dolay\i s\i yla
\begin{equation}\label{eqn}
\text{\colorbox{\ydtfcolor}{\Large{%
$\frac{y^2}{d^2}=\frac{(a+x)^2-a^2}{(a+c)^2-a^2}$.}}}
\end{equation}

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\end{minipage}
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\hfill\raisebox{-800pt}{\fbox{\begin{minipage}[t]{\Bwidth}
\raggedright
%\footnotesize
\small
%\c Simdi
\begin{gather}\label{eqn:prop}
\text{\colorbox{\befcolor}{\color{white}$\bm{EK:EB::EB:EF}$\textbf,}}\\\notag
\text{\colorbox{\befcolor}{\color{white}$\bm{BL\parallel KA}$\textbf,}}\\\notag
\text{\colorbox{\befcolor}{\color{white}$\bm{EA=b,\qquad
AL=e,\qquad
LB=f}$}}
\end{gather}
olsun.  \eqref{eqn:prop} orant\i s\i ndan,
$EF=a+c$ oldu\u gundan
\begin{equation*}
  EK=\frac{a^2}{a+c};
\end{equation*}
\colorbox{\befcolor}{\color{white}$\bm{\triangle AKE\sim\triangle LBE}$} 
benzerli\u ginden
\begin{gather}\label{eqn:AK}
AK=\frac{bf}{b+e},\qquad
AK=\frac{a^2}{a+c}\cdot\frac af=\frac{af}{a+c},\\\label{eqn:KF}
KF=a+c-\frac{a^2}{a+c}=\frac{(a+c)^2-a^2}{a+c}.
\end{gather}
\eqref{eqn:AK} denklemlerinden
\begin{equation}\label{eqn:ba}
  \frac ba=\frac{b+e}{a+c}.
\end{equation}
\begin{equation*}
\text{\colorbox{\xystcolor}{\color{white}$\bm{GM\parallel KA,\qquad
AM=s,\qquad MG=t}$}}
\end{equation*}
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olsun. 
\eqref{eqn} denklemi, bize $x$ ve $y$ aras\i ndaki ili\c skiyi verir;
$s$ ve $t$ aras\i ndaki benzer ili\c skiyi bulaca\u g\i z.
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\footnotesize
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\hfill
\begin{minipage}[t]{\Cwidth}%\tiny%
\footnotesize
Yeni denklemi bulmak i\c cin
\begin{equation*}
  \text{\colorbox{\MNcolor}{\color{white}$\bm{MN\parallel BH}$\textbf,}}\qquad
\text{\colorbox{\MPcolor}{\color{white}$\bm{MP\parallel AF}$}}
\end{equation*}
olsun;
o zaman
\begin{gather}\label{eqn:a+x}
  a+x=EH=EP+PH=EP+MN,\\\label{eqn:y}
y=HG=HN+NG=PM+NG.
\end{gather}
Ayr\i ca 
\colorbox{\MPcolor}{\color{white}$\bm{\triangle AEF\sim\triangle MEP}$}\\
oldu\u gundan

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\begin{gather}\label{eqn:EP}
EP:EM::EF:EA,\qquad EP=\frac{a+c}b\cdot(b+s),\\\label{eqn:PM}
PM:EM::FA:EA,\qquad PM=\frac db\cdot(b+s);
\end{gather}
\colorbox{\MNcolor}{\color{white}$\bm{\triangle AKF\sim\triangle GMN}$} 
oldu\u gundan,\\
\eqref{eqn:AK} ve \eqref{eqn:KF} denklemlerinden

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\begin{gather}\label{eqn:MN}
MN:MG::KF:KA,\qquad MN=\frac{(a+c)^2-a^2}{af}\cdot t,\\\label{eqn:NG}
NG:MG::FA:KA,\qquad NG=\frac{d\cdot(a+c)}{af}\cdot t.
\end{gather}
\eqref{eqn:a+x}, \eqref{eqn:EP}, ve \eqref{eqn:MN} e\c sitliklerine g\"ore
\begin{equation}\label{eqn:a+x2}
a+x=\frac{a+c}b\cdot(b+s)+\frac{(a+c)^2-a^2}{af}\cdot t,
\end{equation}
ve \eqref{eqn:y}, \eqref{eqn:PM}, ve \eqref{eqn:NG} e\c sitliklerine g\"ore
\begin{equation}\label{eqn:y2}
  y=\frac db\cdot(b+s)+\frac{d\cdot(a+c)}{af}\cdot t.
\end{equation}
\c Simdi \eqref{eqn:a+x2} e\c sili\u ginden
\begin{equation*}
\text{\colorbox{\compcolor}{$
\begin{aligned}
\frac{(a+x)^2-a^2}{(a+c)^2-a^2}
&=\frac1{(a+c)^2-a^2}\cdot{}\\
&\phantom{{}={}}{}\cdot\left(\left(\frac{a+c}b\cdot(b+s)+\frac{(a+c)^2-a^2}{af}\cdot t\right)^2
-a^2\right)\\
&=\frac{(a+c)^2}{(a+c)^2-a^2}\cdot\left(\frac{b+s}b\right)^2
+\frac{(a+c)^2-a^2}{a^2f^2}\cdot t^2+{}\\
&\phantom{{}={}}{}+2\cdot\frac{a+c}{abf}\cdot(b+s)\cdot t
-\frac{a^2}{(a+c)^2-a^2},
\end{aligned}$}}
\end{equation*}
ve \eqref{eqn:y2} e\c sitli\u ginden
\begin{equation*}
\text{\colorbox{\compcolor}{$
\begin{aligned}
  \frac{y^2}{d^2}
&=\frac1{d^2}\cdot
\left(\frac db\cdot(b+s)+\frac{d\cdot(a+c)}{af}\cdot t\right)^2\\
&=\left(\frac{b+s}b\right)^2+\frac{(a+c)^2}{a^2f^2}\cdot t^2
+2\cdot\frac{a+c}{abf}\cdot(b+s)\cdot t.
\end{aligned}$}}
\end{equation*}
Bunlar\i\ \eqref{eqn} denklemine koyarak
ve basitle\c stirerek
%\c Simdi yukar\i daki \eqref{eqn} denkleminden
\begin{multline*}
  \left(\frac{b+s}b\right)^2+\frac{(a+c)^2}{a^2f^2}\cdot t^2\\
=\frac{(a+c)^2}{(a+c)^2-a^2}\cdot\left(\frac{b+s}b\right)^2
+\frac{(a+c)^2-a^2}{a^2f^2}\cdot t^2
-\frac{a^2}{(a+c)^2-a^2};
\end{multline*}
benzer terimleri bir araya getirerek
%ve sonu\c c olarak
\begin{align*}
  \frac{t^2}{f^2}
&=\frac{a^2}{(a+c)^2-a^2}\cdot\left(\frac{b+s}b\right)^2
-\frac{a^2}{(a+c)^2-a^2}\\
&=\frac{a^2}{b^2}\cdot\frac{(b+s)^2-b^2}{(a+c)^2-a^2}.
\end{align*}
Son olarak \eqref{eqn:ba} e\c sitli\u ginden
\begin{equation*}
\text{\colorbox{\ydtfcolor}{\Large%
$\frac{t^2}{f^2}=\frac{(b+s)^2-b^2}{(b+e)^2-b^2}$.}}
\end{equation*}
\end{minipage}
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\nopagebreak

\color{blue}
%\vfill
\tiny % \hfill \hfill \hfill
David Pierce\hfill
\url{dpierce@msgsu.edu.tr}\hfill
\url{http://mat.msgsu.edu.tr/~dpierce/}\hfill
Matematik B\"ol\"um\"u\hfill
Mimar Sinan G\"uzel Sanatlar \"Universitesi\hfill
\.Istanbul%\hfill\mbox{} 

%\layout

\end{document}
