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\begin{document}
%\titlehead{\centering}
\title{Analitik Geometri}
\author{David Pierce\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
%  Istanbul\\
  \url{mat.msgsu.edu.tr/~dpierce}\\
\url{polytropy.com}}
\date{13 Ocak 2020}
\maketitle

\section{Giri\c s}

\textbf{Analitik Geometri,}
cebirsel y\"ontemleri kullanan geometridir.
Bu y\"ontemler,
bir \emph{orant\i lar} kuram\i na ba\u gl\i d\i r.
Bu kuramda
\begin{enumerate}[1)]
\item
  orant\i l\i l\i k ba\u g\i nt\i s\i,
  bir denklik ba\u g\i nt\i s\i d\i r;
%  \"ozellikle ge\c ci\c slidir;
\item
  \emph{Thales Teoremi} do\u grudur.
\end{enumerate}
Bu notlarda
\begin{itemize}
\item 
orant\i l\i l\i k kuram\i n\i,
\emph{Desargues Teoremi}'nden;
\item
  Desargues Teoremi'ni,
  \emph{Pappus Teoremi}'nden;
\item
  Pappus Teoremi'ni,
  \"Oklid'in \emph{\"O\u geler}'inin \textsc i.\ kitab\i ndan
\end{itemize}
elde edece\u giz.

\section{Thales Teoremi}

Geometrik tan\i mlardan birine g\"ore bir \textbf{elips,}
iki verilen \textbf{odak noktas\i ndan}
uzakl\i klar\i n\i n toplam\i\ ayn\i\ olan noktalar\i n bir yeridir.
Analitik geometride,
bir $a$ ve bir $b$ i\c cin,
elipsi tan\i mlayan ko\c sul
iki de\u gi\c skeni olan
\begin{equation*}
  \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
\end{equation*}
cebirsel denklemi taraf\i ndan
ifade edilebilir.

Cebirsel bir denklemde
de\u gi\c skenlerin ve sabitlerin de\u gerleri,
ger\c cel say\i lar\i n olu\c sturdu\u gu
$\R$ gibi bir \emph{cisimden} gelir.
Bir cisimde iki eleman\i n
\emph{toplam\i} ve \emph{\c carp\i m\i} vard\i r.

\"Oklid'in \emph{\"O\u geler}'inin birinci kitab\i nda
toplaman\i n geometrik anlam\i\ vard\i r,
ama \c carpma kavram\i\ yoktur
\cite{bones,MR1932864,Oklid-2015}.

1637 y\i l\i nda yay\i nlanm\i\c s olan
\emph{La G\'eometrie}
adl\i\ kitab\i nda
Descartes, ger\c cel say\i lar\i n \c carpmas\i na
geometrik bir anlam\i\ verir
\cite{Descartes-Geometry,Descartes-Geometrie}.
Bunun i\c cin \textbf{Thales Teoremi'ni}
kullan\i r
\cite{Thales-Theorem}.
Bu teoreme g\"ore e\u ger \Sekilde{fig:Thales}ki gibi
\begin{figure}
              \psset{PointSymbol=none}
  \subfloat[]{
    \begin{pspicture}(5,4)
      \pstGeonode[PosAngle={180,180,0}]
        (0.5,0.2)A(0.5,3.8)B(4.5,1.5)O
      \pstHomO[HomCoef=0.6,PosAngle={-90,90}]O{A,B}[C,D]
      \ncline AO\ncline BO
%      \psset{linewidth=2.4pt}
      \ncline AB\ncline CD
  \end{pspicture}}
  \hfill
    \subfloat[]{
    \begin{pspicture}(5,4)
      \pstGeonode[PosAngle={180,180,0,0}]
      (0.5,0.2)A(0.5,3.8)B(4.5,3)C(4.5,1)D
      \pstInterLL[PosAngle=90] ACBDO
      \ncline AC\ncline BD
%      \psset{linewidth=2.4pt}
      \ncline AB\ncline CD
    \end{pspicture}}
\caption{Thales Teoremi}
  \label{fig:Thales}
\end{figure}
bir $OAB$ \"u\c cgeninin $OA$ kenar\i nda $C$
ve $OB$ kenar\i nda $D$ oturursa,
o zaman
\begin{equation*}
  AB\parallel CD\liff\rat{\vec OA}{\vec OC}\prop\rat{\vec OB}{\vec OD}.
\end{equation*}
Burada
\begin{itemize}
\item
  $\vec OA$, bir \emph{y\"onl\"u do\u gru par\c cas\i d\i r};
  \item
    $\rat{\vec OA}{\vec OC}$, bir \emph{orand\i r};
    \item
$\rat{\vec OA}{\vec OC}\prop\rat{\vec OB}{\vec OD}$,
      bir \emph{orant\i d\i r};
    \item
      $\prop$ i\c sareti,
      orant\i l\i l\i k ba\u g\i nt\i s\i n\i\
      (yani oranlar ayn\i l\i\u g\i n\i) g\"osterir.
\end{itemize}
%\"Oklid ve Descartes'\i n do\u gru par\c calar\i, y\"onl\"u de\u gildir.
Thales Teoremi, \emph{\"O\u geler}'in alt\i nc\i\ kitab\i n\i n
ikinci \"onermesidir.
\"Onermeyi g\"ostermek i\c cin \"Oklid,
be\c sinci kitapta bulunan orant\i lar kuram\i n\i\ kullan\i r.
Bu kuram,
uzunluklar\i n
\textbf{Ar\c simet \"ozelli\u gini} varsay\i r.
Bu \"ozelli\u ge g\"ore
iki uzunlu\u gun
daha k\"u\c c\"u\u g\"un\"un bir kat\i,
daha b\"uy\"u\u g\"unden daha b\"uy\"ukt\"ur.

Pozitif ger\c cel say\i lar\i n ayn\i\ \"ozelli\u gi vard\i r.
\.Iki pozitif ger\c el say\i n\i n b\"ol\"um\"u vard\i r,
ve bu b\"ol\"um, pozitif
ger\c cel bir say\i d\i r.
\"Oklid'de iki uzunlu\u gun oran\i\ vard\i r,
ve bu oran, pozitif ger\c cel bir say\i\ olarak anla\c s\i labilir.

\begin{sloppypar}
  Asl\i nda analitik geometri yapmak i\c cin,
  Ar\c simet \"ozelli\u gine ihtiyac\i m\i z yoktur.
  Ayr\i ca oranlar negatif olabilir.
  Orant\i lar\i n \emph{tan\i m\i} olarak
  Thales Teoremi'ni varsayabiliriz.
  Bu durumda orant\i l\i l\i \u g\i n
  ge\c ci\c sli oldu\u gunu g\"ostermek zorunday\i z.
  Bunun i\c cin,
  Hilbert
  1899 y\i l\i nda g\"osterdi\u gi gibi
  \begin{itemize}
  \item 
    4.\ y\"uzy\i l\i nda yay\i nlanm\i\c s olan
    \emph{Pappus Teoremi}
    \cite{Jones-whole,Pappus-Pierce-2},
  \item
    1648 y\i l\i nda yay\i nlanm\i\c s olan \emph{Desargues Teoremi}
    \cite{Desargues-Thm}
  \end{itemize}
  yeter
  \cite{MR0116216}.
  Hessenberg,
  Desargues Teoremi'ni
  Pappus Teoremi'nden
  1905 y\i l\i nda
  elde eder
  \cite{MR1511339}.
  Artin,
  analitik geometri i\c cin
  Pappus ve Desargues Teoremleri'nin yetti\u gini
  Hilbert'inkinden (ve bizimkinden)
  ba\c ska bir \c sekilde
  1957 y\i l\i nda
  g\"osterir
  \cite{Artin}.
  (Hilbert ve Hessenberg'de Pappus Teoremi'ne
  \emph{Pascal Teoremi} denir.)
\end{sloppypar}

\section{Desargues Teoremi: Bildirme}

\"U\c ctane $AD$, $BE$, ve $CF$ do\u grusu
\Sekilde{fig:Des}ki gibi
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      \pstGeonode[PosAngle={180,90,-90,90}]
        (0.5,1.2)A(2,3.4)B(1,0.6)C(4.8,3.4){E}
      \pstTranslation[PosAngle={135,-90}] B{E}{A,C}[D,F]
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      \psset{linewidth=2pt}
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      \ncline AC\ncline{D}{F}
      \psset{linestyle=dashed}
      \ncline BC\ncline{E}{F}
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  \hfill
  \subfloat[]{\label{fig:Des-2}
    \begin{pspicture}(1,0)(6,4)
      \pstGeonode[PosAngle={180,90,-90,0}]
        (0.5,1.2)A(2,3.4)B(1,0.6)C(5.5,1.5)O
      \pstHomO[HomCoef=0.55,PosAngle={135,90,-90}]O{A,B,C}[D,E,F]
      \ncline AO\ncline BO\ncline CO
      \psset{linewidth=2.4pt}
      \ncline AB\ncline{D}{E}
      \ncline AC\ncline{D}{F}
      \psset{linestyle=dashed}
      \ncline BC\ncline{E}{F}
    \end{pspicture}}
  \caption{Desargues Teoremi}
  \label{fig:Des}
\end{figure}
\begin{enumerate}[(a)]
\item
  ya birbirine paralel olsun,
\item
  ya da $O$ noktas\i nda kesi\c ssin.
\end{enumerate}
Ayr\i ca
\begin{align*}
  AB&\parallel DE,&AC&\parallel DF
\end{align*}
olsun.
\textbf{Desargues Teoremi'ne}
g\"ore
\begin{equation*}
  BC\parallel EF.
\end{equation*}
Bu teoremin birinci durumunu elde etmek i\c cin,
\"Oklid'in \emph{\"O\u geler}'inin
birinci kitab\i ndan
\"Onermeler 34 ve 30, birinci ortak kavram,
ve \"Onerme 33 yeter.

\section{Pappus Teoremi: Bildirme ve G\"osterme}

Desargues Teoremi'nin ikinci durumunu
g\"ostermek i\c cin,
\c Sekil \ref{fig:Pappus} veya \numarada{fig:Pappus-duz}ki gibi
\begin{figure}
  \centering
    \begin{pspicture}(10,6)
            \psset{PointSymbol=none}
            %     \psgrid
    \pstGeonode[PosAngle={0,0,90,-90}]
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    \caption{Pappus Teoremi}\label{fig:Pappus}
\end{figure}
$A$ noktas\i\ $EF$'de,
$D$ noktas\i\ $BC$'de olsun,
ve tekrar
\begin{align*}
  AB&\parallel DE,&AC&\parallel DF
\end{align*}
olsun.
\textbf{Pappus Teoremi'ne}
g\"ore
\begin{equation*}
  BF\parallel CE.
\end{equation*}
Bunu kan\i tlamak i\c cin
\Sekilde{fig:Pappus-duz}ki gibi
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%            \ncline{\TF}{\TC}\ncline{\TB}{\TE}\ncline{\TD}{\TA}
            \psset{linestyle=solid}
            \psset{linewidth=2.4pt}
            \ncline{\TA}{\TB}
            \ncline {\TC}{\TA}
            \ncline{\TD}{\TE}      
            \ncline {\TF}{\TD}
            \psset{linestyle=dashed}
            \ncline {\TC}{\TE}
            \ncline {\TB}{\TF}
                    \end{pspicture}

          \caption{Pappus Teoremi'nin d\"uzenlemesi}
          \label{fig:Pappus-duz}
          
        \end{figure}
  ${\TA}{\TD}$, ${\TB}{\TE}$, ve
${\TC}{\TF}$
\c cizilsin.
\"Oklid'in \"Onerme 37'sine g\"ore
  ${\TA}{\TB}\parallel {\TD}{\TE}$ oldu\u gundan
\begin{equation*}
  {\TA}{\TB}{\TD}={\TA}{\TB}{\TE}.
\end{equation*}
\.Ikinci ortak kavrama g\"ore
\begin{equation*}
  {\TF}{\TB}{\TD}{\TA}={\TA}{\TF}{\TB}+{\TA}{\TB}{\TD}
  ={\TA}{\TF}{\TB}+{\TA}{\TB}{\TE}
  ={\TF}{\TB}{\TE}.
\end{equation*}
Benzer \c sekilde
${\TF}{\TD}\parallel {\TA}{\TC}$ oldu\u gundan
\begin{gather*}
  {\TF}{\TD}{\TA}={\TF}{\TD}{\TC},\\
  {\TF}{\TB}{\TD}{\TA}
  ={\TF}{\TB}{\TD}+{\TF}{\TD}{\TA}={\TF}{\TB}{\TD}+{\TF}{\TD}{\TC}={\TF}{\TB}{\TC},
\end{gather*}
dolay\i s\i yla
\begin{equation*}
  {\TF}{\TB}{\TE}={\TF}{\TB}{\TC}.
\end{equation*}
\"Onerme 39'a g\"ore ${\TB}{\TF}\parallel {\TC}{\TE}$.
Pappus'un Teoremi kan\i tlanm\i\c st\i r.

Pappus'un Teoremi'ne g\"ore
iki do\u grunun birinden di\u gerine
bir alt\i gen\i n her kenar\i\ ge\c cerse,
ve iki durumda alt\i genin kar\c s\i t kenarlar\i\
birbirine paralel ise,
o zaman \"u\c c\"unc\"u durumda da paralellerdir.
\c Sekiller \ref{fig:Pappus} ve \numarada{fig:Pappus-duz}
alt\i gen $ABFDEC$ olur.

\section{Desargues Teoremi: G\"osterme}

\c Simdi
%Desargues Teoremi'nin ikinci durumunu kan\i tlamak i\c cin,
  \Sekilde{fig:Hessen}ki gibi
\begin{figure}
  \centering
    \psset{PointSymbol=none}
\subfloat[]{  \begin{pspicture}(10,7)
%    \psgrid[subgriddiv=1]
    \pstGeonode[PosAngle={180,180,-90,90}]
    (0.5,6.8)L(0.5,0.2)C(9.8,1)O(9.8,5.5)N
    \pstHomO[HomCoef=0.55,PosAngle=180]CL[A]
    \pstHomO[HomCoef=0.45,PosAngle=90]NL[B]
    \pstTranslation[PointName=none] ABN[x]
    \pstInterLL[PosAngle=-150]NxOA{D}
    \pstInterLL[PosAngle=75]NxOB{E}
    \pstTranslation[PointName=none] AC{D}[x]
    \pstInterLL[PosAngle=-90]{D}xOC{F}
    \pstInterLL[PosAngle=-90] L{D}OCM
    {\psset{linewidth=0pt,hatchangle=0}
    \pspolygon[fillstyle=solid,fillcolor=lightgray]
    (O)(N)(M)(L)(C)(B)
%    \pspolygon[fillstyle=solid,fillcolor=gray]
    \pspolygon[fillstyle=hlines]
    (O)(N)(M)(D)(F)(E)
    \pspolygon[fillstyle=vlines](O)(N)(D)(L)(A)(B)}
%    \uput*[90]({D}){$D$}
    \ncline OA\ncline OC
%    \ncline OB
%    \ncline ML
    \ncline LN
    \ncline NO
    \psset{linewidth=3pt}
    \ncline LC\ncline{D}{F}
    \ncline AB\ncline{D}{N}
    \psset{linestyle=dashed}
    \ncline BC\ncline{E}{F}    
    \ncline NM
\end{pspicture}}

\subfloat[]{\begin{pspicture}(10,5)
    \pstGeonode[PosAngle={-90,90,90}](0.2,0.6)C(1,4.4)A(9.8,4.4)B
    \pstTranslation ABC[O]
    \pstHomO[HomCoef=0.7,PosAngle={90,0,-90}]O{A,B,C}[D,G,F]
    \ncline OA\ncline OB\ncline OC
    \ncline AB
    \ncline AC\ncline DF
    \ncline BC\ncline{G}F
    \psset{linestyle=dotted}
    \ncline D{G}
  \end{pspicture}}
  
  \caption{Desargues i\c cin Hessenberg'in kan\i t\i}
  \label{fig:Hessen}
\end{figure}
$AD$, $BE$, ve $CF$'nin ortak noktas\i\ $O$ olsun;
ayr\i ca $AB\parallel DE$ ve $AC\parallel DF$ olsun.
\.Iki durum vard\i r.
\begin{enumerate}[(a)]
\item
  E\u ger $BACO$ bir paralelkenar de\u gilse,
  $OB\nparallel AC$ varsay\i labilir.
  Bu durumda
$LM\parallel OB$ ve $D\in LM$ olsun.
  Pappus Teoremi ile
\begin{itemize}
\item
  $ONDLAB$ alt\i geninde $ON\parallel AL$,
\item
  $ONMLCB$ alt\i geninde $BC\parallel MN$,
\item
  $ONMDFE$ alt\i geninde $EF\parallel MN$.
\end{itemize}
Paralellik ge\c ci\c sli oldu\u gundan
$BC\parallel EF$.
\item
  E\u ger $BACO$ bir paralelkenar ise,
  o zaman $ABCO$ bir paralelkenar de\u gildir.
  Bu durumda
  $OB$'de bir $G$ i\c cin $FG\parallel CB$.
  \.Ilk durumdan $DG\parallel AB$,
  dolay\i s\i yla $G$ ve $E$ noktas\i\ ayn\i d\i r.
  \qedhere
\end{enumerate}
Desargues Teoremi'nin ikinci durumu kan\i tlanm\i\c st\i r.

\section{Vekt\"orler Grubu}

\"Oklid'de do\u gru par\c calar\i n e\c sitli\u gi
bir denklik ba\u g\i nt\i s\i d\i r.
\"Ozellikle birinci ortak kavrama g\"ore
e\c sitlik ge\c ci\c slidir.
E\c sitli\u ge g\"ore bir do\u gru par\c cas\i n\i n denklik s\i n\i f\i,
%do\u gru par\c cas\i n\i n
onun
\textbf{uzunlu\u gu} olarak
anla\c s\i labilir.

E\u ger bir do\u gru kendisine paralel ise,
o zaman \"Onerme 30'a g\"ore paralellik de
bir denklik ba\u g\i nt\i s\i d\i r.

Desargues Teoremi'nin
ilk durumu sayesinde
y\"onl\"u do\u gru par\c calar\i n\i n,
a\c sa\u g\i daki ko\c sullar\i\ sa\u glayan
\emph{e\c sitli\u gi} vard\i r,
ve bu e\c sitlik,
bir denklik ba\u g\i nt\i s\i d\i r.
\begin{enumerate}
\item 
  Herhangi $ABDC$ paralelkenari i\c cin
  o zaman
  \begin{equation*}
    \vec AB=\vec CD.
  \end{equation*}
  \item
    Herhangi $\vec AB$ y\"onl\"u do\u gru par\c cas\i\ ve
    $C$ noktas\i\ i\c cin,
  bir ve tek bir $D$ noktas\i\ i\c cin,
  \begin{equation*}
    \vec AB=\vec CD.
  \end{equation*}
\end{enumerate}
E\c sitli\u ge g\"ore
y\"onl\"u bir do\u gru par\c cas\i n\i n
denklik s\i n\i f\i,
bir \textbf{vekt\"ord\"ur.}

Bir vekt\"or daha vard\i r.
$\vec AA$'n\i n y\"onl\"u do\u gru par\c cas\i\ olmad\i\u g\i\ halde
her durumda
\begin{equation*}
  \vec AA=\vec BB
\end{equation*}
olsun, ve her $\vec AA$'n\i n temsil etti\u gi vekt\"or,
\begin{equation*}
  \bm0
\end{equation*}
olsun.
Ayr\i ca
tan\i ma g\"ore
\begin{gather*}
  \vec AB+\vec BC=\vec AC,\\
  -\vec AB=  \vec BA
\end{gather*}
olsun.
Bu tan\i mlar, vekt\"orlere ge\c cer,
ve herhangi $\bm a$, $\bm b$, ve $\bm c$ vekt\"or\"u i\c cin
\begin{gather*}
  (\bm a+\bm b)+\bm c=\bm a+(\bm b+\bm c),\\
  \bm b+\bm a=\bm a+\bm b,\\
  \bm a+\bm0=\bm a,\\
  \bm a+(-\bm a)=\bm0.
\end{gather*}
B\"oylece vekt\"orler
$V$ k\"umesini olu\c sturdu\u gunda
$(V,+,-,\bm0)$ bir \textbf{abelyan gruptur.}

\section{Oranlar Cismi}

Desargues Teoremi'nin
ikinci durumu sayesinde,
iki paralel y\"onl\"u do\u gru par\c cas\i n\i n
\textbf{oran\i} vard\i r,
ve ayr\i ca onlar\i n temsil etti\u gi
vekt\"orlerin ayn\i\ oran\i\ vard\i r.
Bir oran, bir denklik s\i n\i f\i d\i r,
ve a\c sa\u g\i daki ko\c sullar sa\u glan\i r.
\begin{enumerate}
\item 
  Thales Teoremi do\u grudur.
\item
  Herhangi $\rat{\vec OA}{\vec OB}$ oran\i\
  ve $\vec OC$ y\"onl\"u do\u gru par\c cas\i\ i\c cin,
  bir ve tek bir $D$ noktas\i\ i\c cin,
  \begin{equation*}
    \rat{\vec OA}{\vec OB}\prop\rat{\vec OC}{\vec OD}.
  \end{equation*}
\end{enumerate}
Bir oran daha vard\i r.
\begin{equation*}
  \rat{\vec OO}{\vec OA}\prop\rat{\vec OO}{\vec OB}
\end{equation*}
olsun, ve ortak oran $0$ olsun.
\c Simdi tan\i ma g\"ore
\begin{gather*}
  \rat{\bm a}{\bm c}+\rat{\bm b}{\bm c}=\rat{(\bm a+\bm b)}{\bm c},\\
  \rat{-\bm a}{\bm b}=-(\rat{\bm a}{\bm b}),\\
  (\rat{\bm a}{\bm b})\cdot(\rat{\bm b}{\bm c})=\rat{\bm a}{\bm c},\\
   \bm b\neq\bm 0\lto\rat{\bm b}{\bm a}=(\rat{\bm a}{\bm b})^{-1},\\
  \rat{\bm a}{\bm a}=1
\end{gather*}
olsun.
Pappus Teoremi sayesinde
\begin{equation*}
  (\rat{\bm a}{\bm b})\cdot(\rat{\bm c}{\bm d})
  \prop
  (\rat{\bm c}{\bm d})\cdot(\rat{\bm a}{\bm b}).
\end{equation*}
B\"oylece
oranlar
$K$ k\"umesini olu\c sturdu\u gunda
\begin{itemize}
\item
  $(K,+,-,0)$ bir abelyan gruptur,
\item
  $(K\setminus\{0\},{}\cdot{},{}^{-1},1)$ bir abelyan gruptur,
\item
  $K$'de
    herhangi $a$, $b$, ve $c$ oran\i\ i\c cin
    \begin{equation*}
      a\cdot(b+c)=a\cdot b+a\cdot c.
    \end{equation*}
\end{itemize}
Bundan dolay\i\
    $(K,{}\cdot{},+,-,1,0)$ bir \textbf{cisimdir.}

\section{Koordinatlar}

\c Simdi tan\i ma g\"ore
\begin{equation*}
  (\rat{\vec OA}{\vec OB})\cdot\vec OB=\vec OA
\end{equation*}
olsun.
O zaman herhangi $a$ ve $b$ oran\i\
ve herhangi $\bm c$ ve $\bm d$ vekt\"or\"u i\c cin
\begin{gather*}
  a\cdot(\bm c+\bm d)=a\cdot\bm c+a\cdot\bm d,\\
  (a+b)\cdot\bm c=a\cdot\bm c+b\cdot\bm c,\\
  a\cdot(b\cdot\bm c)=(a\cdot b)\cdot\bm c,\\
  1\cdot\bm c=\bm c.
\end{gather*}
B\"oylece vekt\"orler grubu,
oranlar cismi alt\i nda bir \textbf{vekt\"or uzay\i d\i r.}
Ayr\i ca $OAB$ bir \"u\c cgen oldu\u gunda
herhangi $P$ noktas\i\ i\c cin,
girdileri $K$'de olan
bir ve tek bir s\i ral\i\ ikilisi
\begin{equation*}
  \vec OP=x\cdot\vec OA+y\cdot\vec OB
\end{equation*}
denklemini sa\u glar.
O girdiler,
$P$'nin
\emph{Descartes koordinatlar\i} veya
\textbf{kartezyan koordinatlar\i d\i r.}

\section{Do\u grular}

Her $CD$ do\u grusu i\c cin,
$K$'nin bir eleman\i n\i n
\begin{equation*}
  \vec OP=\vec OC+t\cdot\vec CD
\end{equation*}
denklemini sa\u glamas\i,
$P$ noktas\i n\i n $CD$'de olmas\i n\i n
gerek ve yeter bir ko\c suldur.
\c Simdi
\begin{align*}
  \vec OC&=c_1\cdot\vec OA+c_2\cdot\vec OB,&
    \vec OD&=d_1\cdot\vec OA+d_2\cdot\vec OB
\end{align*}
olsun.
O zaman
\begin{multline*}
  \vec OC+t\cdot\vec CD\\
  \begin{aligned}
  &=\vec OC+t\cdot(\vec CO+\vec OD)\\
  &=(1-t)\cdot\vec OC+t\cdot\vec OD\\
  &=(1-t)\cdot(c_1\cdot\vec OA+c_2\cdot\vec OB)
  +t\cdot(d_1\cdot\vec OA+d_2\cdot\vec OB)\\
  &=\bigl((1-t)c_1+td_1\bigr)\cdot\vec OA
  +\bigl((1-t)c_2+td_2\bigr)\cdot\vec OB\\
  &=\bigl(c_1+t(d_1-c_1)\bigr)\cdot\vec OA
  +\bigl(c_2+t(d_2-c_2)\bigr)\cdot\vec OB.
\end{aligned}
\end{multline*}
B\"oylece
\begin{multline*}
  \vec OC+t\cdot\vec CD
  =x\cdot\vec OA+y\cdot\vec OB\\
  \liff
  x-c_1=t(d_1-c_1)\land y-c_2=t(d_2-c_2),
\end{multline*}
dolay\i s\i yla
bir $P$ noktas\i n\i n kartezyan koordinatlar\i n\i n
\begin{equation}\label{eqn:d_2-c_2}
  (d_2-c_2)(x-c_1)=(d_1-c_1)(y-c_2)
\end{equation}
denklemini sa\u glamas\i,
$P$'nin $CD$'de olmas\i n\i n
gerek ve yeter bir ko\c suldur.
O denklem
\begin{equation*}
  (d_2-c_2)x+(c_1-d_1)y+c_2d_1-c_1d_2=0
\end{equation*}
bi\c ciminde yaz\i labilir.

Katsay\i lar\i\ $K$'den gelen,
$a$ ve $b$'nin en az birinin $0$'dan farkl\i\ oldu\u gu
herhangi
\begin{equation}\label{eqn:abc}
  ax+by+c=0
\end{equation}
denkleminin bir $(c_1,c_2)$ \c c\"oz\"um\"u vard\i r.
Bu durumda
denklem
\begin{equation*}
a(x-c_1)=-b(y-c_2)
\end{equation*}
bi\c ciminde yaz\i labilir.
O zaman
\begin{align*}
  d_1&=-b+c_1,&d_2&=a+c_2
\end{align*}
olmak \"uzere $(d_1,d_2)$ de denklemin bir \c c\"oz\"um\"ud\"ur.
Bu durumda denklem, \eqref{eqn:d_2-c_2}
bi\c ciminde yaz\i labilir.

Bir $K$ cismi verilirse,
$a$ ve $b$'nin en az birinin $0$'dan farkl\i\ oldu\u gu
\eqref{eqn:abc} denklemleri,
$K^2$ kartezyan \c carp\i m\i n\i n do\u grular\i n\i\ tan\i mlas\i n.
O zaman Pappus ve Desargues Teoremleri do\u grudur,
ve yukar\i daki gibi elde edilen oranlar,
$K$ cismini olu\c sturur.


\AfterBibliographyPreamble{\relscale{0.9}}
%\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{$'$}
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\end{thebibliography}


\end{document}
