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\newtheorem*{solution}{\c C\"oz\"um}

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\begin{document}

\title{\"Oklid Geometrisine Giri\c s}
\subtitle{S\i nav \c c\"oz\"umleri}
\date{27 Kas\i m 2018}
\author{Sevan Bedikyan \and David Pierce \and G\"ulay Telsiz \and \.Ipek Tuvay}
\maketitle
\thispagestyle{empty}

%\noindent\rput[tl](0,8){Ad Soyad:}%
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\begin{problem}
  A\c sa\u g\i daki kelimelerin T\"urk\c cesi nedir?
  \begin{center}\relscale{0.9}
    \gr{GKALNTIRIMI}\hfill\gr{PANTZARI}\hfill\gr{REKLAMA}\hfill\gr{FILOSOFIA}\hfill\gr{QAMAM}
  \end{center}

\end{problem}

\begin{solution}
    Kald\i r\i m, pancar, reklam, felsefe, hamam.
\end{solution}



\begin{problem}\label{dik}
  A\c sa\u g\i daki \"onermenin d\"uzenlemesini (varsa)
  ve g\"ostermesini yaz\i n.
  \begin{description}
\item[A\c c\i klama]
  Verilmi\c s \"u\c cgenler \gr{ABG} ve \gr{DEZ} olsun, ve
  \begin{align*}
    \grm{AB}&=\grm{DE},&
    \grm{BG}&=\grm{EZ}
  \end{align*}
  olsun, ve \gr A ve \gr D'daki a\c c\i lar dik olsun.
\item[Belirtme]
  Diyoruz ki $\grm{AG}=\grm{DZ}$.
\end{description}
\end{problem}

\begin{solution}\mbox{}
  \begin{description}
    \item[D\"uzenleme]
      M\"umk\"unse  $\grm{AG}<\grm{DZ}$ olsun.
      O zaman
  \begin{enumerate}
    \item
  \gr{DZ}'dan \gr{AG}'ya e\c sit olan \gr{DH} kesilsin
     [\"On.\ 3];
   \item
     \gr{EH} \c cizilsin [P.\ 1].
  \end{enumerate}
   \item[G\"osterme]\mbox{}
     \begin{enumerate}\setcounter{enumi}2
       \item
         $\grm{EH}=\grm{BG}$ [P.\ 4 ve \"On.\ 4].
         \item
           Dolay\i s\i yla $\grm{HE}=\grm{EZ}$ [P.\ 1].
           \item
             Bu durumda $\grm{EHZ}=\grm{EZH}$ [\"On.\ 5].
             \item
               \gr{EHZ}, dik a\c c\i dan b\"uy\"ukt\"ur [\"On. 16].
               \item
                 \c Simdi \"On. 17 ile bir \c celi\c ski vard\i r.
  \end{enumerate}
  \end{description}
\end{solution}


\begin{problem}\mbox{}
  \begin{enumerate}[a)]
    \item
      A\c sa\u g\i daki \"onermenin her numaral\i\ ad\i m\i\ i\c cin,
      kulland\i\u g\i\ postulat, ortak kavram, veya \"onerme varsa,
      numaras\i n\i\ yaz\i n.
      Bu ad\i mlardan birinde bir \"onceki problem de kullan\i labilir.
      \.Ilk ad\i m size \"ornek olarak verilmi\c stir.
%      Problem 1 de kullan\i labilir.
%      Bir \"ornek veriliyor.
    \item
      \"Onerme do\u gru mu, yanl\i\c s m\i?
A\c sa\u g\i da a\c c\i klay\i n.
  \end{enumerate}
      \begin{description}
      \item[Bildirme]
        Her \"u\c cgen ikizkenard\i r.
\item[A\c c\i klama]
  Verilmi\c s \"u\c cgen \gr{ABG} olsun.
\item[Belirtme]
  Diyoruz ki
  $\grm{AB}=\grm{AG}$.
\item[D\"uzenleme]\mbox{}
  \begin{enumerate}[1.]
    \item
      \gr{BAG}'n\i n \gr{AD} a\c c\i ortay\i\ \c cizilsin.  [\"On.\ 9]
    \item
      \gr{BG}'n\i n \gr E orta noktas\i\ belirtilsin.
    \item
      \gr{BG}'n\i n \gr{EZ} dikmesi \c cizilsin.
    \item
      \gr{AD} ve \gr{EZ}, \gr H noktas\i nda kesi\c ssin.
    \item
      Gerekirse \gr{AB} ve \gr{AG} uzat\i ls\i n.
    \item
      \gr{AB}'ya veya uzat\i lmas\i na \gr{HJ} dikmesi indirilsin.
    \item
      \gr{AG}'ya veya uzat\i lmas\i na \gr{HK} dikmesi indirilsin.
  \end{enumerate}
\item[G\"osterme]\mbox{}
  \begin{enumerate}[1.]\setcounter{enumii}7
  \item
    $\grm{AJH}=\grm{AKH}$.
  \item
    $\grm{AJ}=\grm{AK}$.
  \item
    $\grm{HJ}=\grm{HK}$.
  \item
    $\grm{BEH}=\grm{GEH}$.
  \item
    $\grm{BH}=\grm{GH}$.
  \item
    $\grm{BJ}=\grm{KG}$.
  \item\label 3
    $\grm{AB}=\grm{AG}$.
  \end{enumerate}
      \item[Bildirme]
        B\"oylece her \"u\c cgen ikizkenard\i r.
\end{description}
\end{problem}

\begin{solution}
  \begin{enumerate}[a)]
  \item
    \begin{minipage}[t]{0.45\textwidth}
D\"uzenleme:
    \begin{enumerate}[1:]
    \item \"On.\ 9.
    \item
      \"On.\ 10.
    \item
      \"On.\ 11.
    \item
      ---
    \item
      P.\ 2.
    \item
      \"On.\ 12.
    \item
      \"On.\ 12.
    \end{enumerate}
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\textwidth}
  G\"osterme
    \begin{enumerate}[1:]
    \item
      P.\ 4.
    \item
      \"On.\ 26.
    \item
      \"On.\ 26.
    \item
      P.\ 4.
    \item
      \"On. 4.
    \item
      Problem \ref{dik}.
      \item
      Yanl\i\c st\i r.
    \end{enumerate}
    \end{minipage}
  \item
  \"Onerme yanl\i\c st\i r.
  G\"ostermenin son ad\i m\i nda
  \begin{align*}
    \grm{AB}&=\grm{AJ}\pm\grm{BJ},&
    \grm{AG}&=\grm{AK}\pm\grm{GK}
  \end{align*}
  varsay\i l\i yor, ama
  $\grm{AG}=\grm{AK}\mp\grm{GK}$ olur.
  \end{enumerate}

\end{solution}



\begin{problem}
    Tan\i ma g\"ore ger\c cel say\i larda
    e\u ger $x-y$ bir tamsay\i\ ise $x\mathrel By$ olsun.
  \begin{enumerate}[a)]
  \item
    $B$  ba\u g\i nt\i s\i n\i n
    bir denklik ba\u g\i nt\i s\i\ oldu\u gunu g\"osterin.
  \item
    $\sqrt 3$ say\i s\i n\i n
    denklik s\i n\i f\i n\i n $4$ tane eleman\i\ yaz\i n.
  \item
    $B$ ba\u g\i nt\i s\i n\i,
    $\R\times\R$ \c carp\i m\i n\i n bir altk\"umesi olarak
    ifade edin.
  \end{enumerate}
\end{problem}

\begin{solution}
    \begin{enumerate}[a)]
    \item
      \begin{itemize}
       \item
         $x-x\in\Z$ \c c\"unk\"u $x-x=0$;
       \item
         $x-y\in\Z$ ise $y-x\in\Z$ \c c\"unk\"u $y-x=-(x-y)$;
         \item
        $x-y\in\Z$ ve $y-z\in\Z$ ise $x-z\in\Z$ \c c\"unk\"u
        $x-z=(x-y)+(y-z)$.
      \end{itemize}
    \item
      $\sqrt3$, $\sqrt3\pm1$, $\sqrt3+2$.
    \item
      $\{(x,y)\in\R^2\colon x-y\in\Z\}$.
    \end{enumerate}
\end{solution}

\end{document}
