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\newtheorem{exercise}{Al\i\c st\i rma}
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\begin{document}
\title{Geometri al\i\c st\i rmalar\i}
\author{Ahmet Bakkalo\u glu \and Ayhan G\"unayd\i n \and \"Ozer \"Ozt\"urk \and David Pierce}
\date{12 Aral\i k 2012}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\url{http://mat.msgsu.edu.tr/}
}

\uppertitleback{
\centering
Bu al\i\c st\i rmalar,\\
 sadece \"Oklid'in \emph{\"O\u geleri}'nin birinci kitab\i n\i n \"onermelerini kullan\i r.
 
Buldu\u gumuz ispatlar,\\
 3--5, 12--19, 23, 26, 28--36, 38--39, 41, ve 43\\
  numaral\i\ \"onermeleri kullan\i yor.

Al\i\c st\i rmalar, 4 b\"ol\"ume ayr\i lm\i\c st\i r.\\
Bir al\i\c st\i rma, b\"ol\"um\"unde kendinden \"once gelen al\i\c st\i rmalar\i\ kullanabilir.

\mbox{}

Bir al\i\c st\i rma anlatacaksan\i z,\\
notlar\i n\i z\i\ kullanmadan,\\
farkl\i\ \c sekiller kullanarak,\\
farkl\i\ harfler kullanarak\\
anlatabilmelisiniz.

Hangi \"onermeleri kulland\i\u g\i n\i z\i\ s\"oylemelisiniz.
}

\lowertitleback{\tableofcontents}

\maketitle


\section{Steiner--Lehmus Teoremi}

\begin{exercise}
\.Ikizkenar \"u\c cgende tabandaki a\c c\i lar\i\ ikiye b\"olenlerin (yani \emph{a\c c\i ortaylar\i n}) birbirine e\c sit old\u gunu g\"osterin.
\end{exercise}

\begin{exercise}
Taban\i ndaki a\c c\i lar\i\ ikiye b\"olenlerin e\c sit oldu\u gu \"u\c cgenin ikizkenar oldu\u gunu g\"osterin.  \emph{\.Ipucu:}  A\c sa\u g\i daki \c sekli kullanarak kar\c s\i t tersini g\"osterin.

\begin{minipage}[b]{.67\textwidth}
$ABC$ \"u\c cgeninde
\begin{compactenum}[1)]
\item
$BD$ ile $CE$, tabandaki a\c c\i lar\i\ ikiye b\"oler;
\item
$FCE$ a\c c\i s\i, $ABD$ a\c c\i s\i na e\c sittir;
\item
$AB$ kenar\i n\i n $BG$ par\c cas\i, $CF$ do\u grusuna e\c sittir;
\item
$BGH$ a\c c\i s\i, $BFC$ a\c c\i s\i na e\c sittir.
\end{compactenum}
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 \end{exercise}

\section{\"U\c cgenlerin merkezleri}

\begin{exercise}\mbox{}\label{KKA}
\.Iki \"u\c cgende, 
tabandaki bir a\c c\i\ tabandaki bir a\c c\i ya e\c sitse,
a\c c\i y\i\ g\"oren kenar a\c c\i y\i\ g\"oren kenara e\c sitse,
ve kalan kenar kalan kenara e\c sitse,
ya tabanlar birbirine e\c sittir,
ya da tabanlardaki kalan a\c c\i lar\i n biri oput, biri dard\i r.
\end{exercise}

\begin{exercise}\mbox{}\nopagebreak

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$ABC$ \"u\c cgeninde
$BD$ ile $CD$, tabandaki a\c c\i lar\i\ ikiye b\"oler.  $AD$ do\u grusunun $BAC$ a\c c\i s\i n\i\ ikiye b\"old\"u\u g\"un\"u g\"osterin.
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\begin{exercise}\mbox{}\nopagebreak\label{circum}

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$ABC$ \"u\c cgeninde
$DE$ do\u grusu, $AB$ kenar\i na dik ve bu kenar\i\ ikiye b\"oler.
Benzer \c sekilde $FE$ do\u grusu, $AC$ kenar\i na dik ve bu kenar\i\ ikiye b\"oler.
$EG$ do\u grusu, $BC$ kenar\i na dikse,
bu kenar\i\ ikiye b\"old\"u\u g\"un\"u g\"osterin.
}
\end{exercise} 

\begin{exercise}\mbox{}

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$ABC$ \"u\c cgeninde
$CD$ do\u grusu, $AB$ kenar\i na diktir, ve $BE$ do\u grusu, $AC$ kenar\i na diktir.
Bu $CD$ ile $BE$ do\u grular\i, $F$ noktas\i nda kesi\c sirler.  
$AG$ do\u grusu, $F$ noktas\i ndan ge\c cer.
$AG$ do\u grusunun $BC$ taban\i na dik oldu\u gunu g\"osterin.
\emph{\.Ipucu:} $HKL$ \"u\c cgeninin kenarlar\i, $ABC$ \"u\c cgeninin kenarlar\i na paralel olsun.
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\end{exercise}

\section{Pisagor Teoreminin Pappus'un verdi\u gi genelle\c stirilmesi}

\begin{exercise}\label{paral}
\.Iki paralelkenarda,
iki biti\c sik kenar, iki biti\c sik kenara e\c sit ise, ve i\c cerilen a\c c\i, i\c cerilen a\c c\i ya e\c sit ise, paralelkenarlar birbirine e\c sittir.
\end{exercise}

\begin{exercise}\mbox{}

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$ABC$, herhangi bir \"u\c cgendir, ve $AD$ ile $AF$, $ABC$ \"u\c
cgeninin kenarlar\i nda rasgele se\c cilmi\c s iki paralelkenard\i r.
Gerekirse, bu paralelkenarlar\i n $DE$ ile $FG$ kenarlar\i\ uzat\i l\i
r, ve $H$ noktas\i nda kesi\c sirler.  $KLMN$ paralelkenar\i nda, $KL$
taban\i, $ABC$ \"u\c cgeninin $BC$ taban\i na e\c sittir; ve $NKL$ a\c
c\i s\i, $ABC$ ile $DHA$ a\c c\i lar\i n\i n toplam\i na e\c sittir.
$AD$ ile $AF$ paralelkenarlar\i n\i n toplam\i n\i n $KLMN$
paralelkenar\i na e\c sit oldu\u gunu g\"osterin.  \emph{\.Ipucu:} 
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\end{exercise}

\section{Pisagor Teoremi i\c cin \"Oklid'in \c sekli}

\begin{exercise}\label{X}
Verilen bir do\u grunun bir noktas\i ndan iki do\u gru, ayr\i\ tarafa \c cizilsin.  \"Oklid'in 15.\ \"onermesine g\"ore, e\u ger bu iki do\u gru, bir do\u grudaysa, o zaman verilen do\u gruyla olu\c sturduklar\i\ ters a\c c\i lar birbirine e\c sittir.  Bu \"onermenin tersini g\"osterin.
\end{exercise}

\begin{exercise}\label{par-diag}
Bir paralelkenar\i n k\"o\c segenlerinin birbirini ikiye b\"ol\-d\"u\u g\-\"un\"u g\"osterin.
\end{exercise}

\begin{exercise}\label{taban}
\.Iki paralelkenar, ayn\i\ paralellerde olsun.  \"Oklid'in 36.\ \"on\-ermesine g\"ore, paralelkenarlar\i n tabanlar\i\ birbirine e\c sitse, paralelkenarlar da birbirine e\c sittir.  Tersini g\"osterin.
\end{exercise}

\begin{exercise}\mbox{}\label{iki}

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$DE$ do\u grusu, $ABC$ \"u\c cgeninin $BC$ taban\i na paraleldir.
$AF$ do\u grusu, taban\i\ ikiye b\"oler.
$AF$ do\u grusunun $DE$ do\u grusunu da ikiye b\"old\"u\u g\"un\"u g\"osterin.
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\begin{exercise}\mbox{}\label{par-div}

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\c Sekilde $DE$ do\u grusu, $BC$ do\u grusuna paraleldir, ve $BD$, $CE$, ve $FG$ do\u grular\i, birbiriyle $A$ noktas\i nda kesi\c sirler.  $FG$ do\u grusu, $BC$ do\u grusunu ikiye b\"oler.  $FG$ do\u grusunun $DE$ do\u grusunu da ikiye b\"old\"u\u g\"un\"u g\"osterin.
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\end{exercise}

\pagebreak

\begin{exercise}\mbox{}\nopagebreak\label{comp}

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$ABCD$ bir paralelkenard\i r, ve $EF$ ile $GH$ do\u grular\i, paralelkenar\i n kenarlar\i na paraleldir.  \"Oklid'in 43.\ \"onermesine g\"ore, e\u ger $AK$ ile $KC$ do\u grular\i, bir do\u grudaysa, o zaman $DK$ ile $KB$ paralelkenarlar\i, birbirine e\c sittir.  Bu \"onermenin tersini g\"osterin.
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\emph{\.Ipucu:} $FG$, $GE$, $EH$, $HF$, ve $AKL$, do\u grular olsun.  O zaman
\begin{compactenum}[1)]
\item
$AL$, $GE$ do\u grusunu ikiye b\"oler;
\item
$EFG$ ile $EHG$ \"u\c cgenleri, birbirine e\c sittir;
\item
$AL$, $FH$ do\u grusunu ikiye b\"oler;
\item
$FLK$ ile $CLH$ a\c c\i lar\i, birbirine e\c sittir.
\end{compactenum}
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\end{exercise}

\begin{exercise}\mbox{}\nopagebreak\label{cut-square}

\begin{minipage}[b]{0.5\textwidth}
\c Seklimiz, \"Oklid'in 47.\ \"onermesinden al\i nm\i\c st\i r.
$AD$ do\u grusu, $BC$ do\u grusuna diktir; $AE$ d\"ortgeni, bir dikd\"ortgendir; ve $AE$ do\u grusu, onun k\"o\c segenidir.  $AD$ ile $AE$ do\u grular\i n\i n bir do\u gruda oldu\u gunu g\"osterin.
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\pagebreak

\begin{exercise}\mbox{}\label{square}

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$EFGH$ paralelkenar\i nda $BK$ ile $LC$ do\u grular\i, kenarlara paraleldirler, ve $A$ noktas\i nda kesi\c sir.
$EL$ paralelkenar\i n\i n $CH$ k\"o\c segeni \c cizilmi\c stir.
  $GA$ do\u grusu \c cizilmi\c s ve $CH$ do\u grusundaki $M$ noktas\i na uzat\i lm\i\c st\i r.  $MB$ ile $MF$ do\u grular\i\ \c cizilmi\c stir.
  Bu $MB$ ile $MF$ do\u grular\i n\i n bir do\u gruda oldu\u gunu g\"osterin.
\end{exercise}

\begin{exercise}\mbox{}\nopagebreak

\parbox[b]{0.5\textwidth}{
\"Oklid'in 47.\ \"onermesinin \c seklinde $AD$, $BE$, ve $CF$ do\u grular\i n\i n bir noktada kesi\c sti\u gini g\"osterin.  (Burada $AD$ do\u grusu, $BC$ do\u grusuna diktir.)
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