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\begin{document}
\title{\"Oklid Geometrisine Giri\c s}
\subtitle{Eski aras\i nav ve final geometri sorular\i}
\author{MSGS\"U Matematik B\"ol\"um\"u}
\date{20 Kas\i m 2018}
\maketitle

%\thispagestyle{empty} 

\begin{problem}
 Verilmi\c s s{\i}n{\i}rlanm{\i}\c s do\u gruda e\c skenar olmayan bir ikizkenar \"u\c cgen in\c sa edin. E\u ger m\"umk\"unse bunu \"Oklid'in \"onermelerini kullanmadan yap{\i}n. 
\end{problem}


\begin{problem}
\"U\c c a\c c{\i}s{\i} e\c sit olan bir \"u\c cgenin e\c skenar olaca\u g{\i}n{\i} ispatlay{\i}n.
\end{problem}


\begin{problem}\sloppy
\gr{K'umh} kolonisi, M.\"O. 8. y\"uzy{\i}lda, bug\"un Napoli olan b\"olgenin yan{\i}nda,  
E\u griboz'dan ve Bat{\i} Anadolu'da \c simdinin Alia\u ga's{\i}%%%%%
\footnote{Paul Harvey, \emph{The Oxford Companion to
    Classical Literature} (1980); Bilge Umar, \emph{T\"urkiye'deki
    Tarihsel Adlar} (\.Istanbul: \.Inkil\^ap, 1993).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
yak{\i}nlar{\i}ndaki \gr{K'umh}'den g\"o\c cenler taraf{\i}ndan
 kurulmu\c stur. Yunan alfabesinin \gr{K'umh} kolonisi'nde kullan{\i}lan halinden t\"uretilen Latin alfabesinin zaman i\c cinde 23 harf{}i olmu\c stur:
 \begin{center}
 A\gap B\gap C\gap D\gap E\gap F\gap G\gap H\gap I\gap K\gap L\gap M\gap N\gap O\gap P\gap Q\gap R\gap S\gap T\gap V\gap X\gap Y\gap Z
 \end{center}
 $863$ y{\i}l{\i}nda, Kiril ad{\i}nda Selanik'li bir rahip, kutsal yaz{\i}tlar{\i} Yunancadan, Eski Bulgarca'ya \c cevirmek i\c cin 
 Glagol denen alfabeyi icat etmi\c stir. Hemen sonra, daha basit olan Kiril alfabesi icat edilmi\c stir.%%%%%
 \footnote{S. H. Gould, 
 \emph{Russian for the Mathematician} (Springer-Verlag, Berlin--Heidelberg--New York, 1972). Pek \c cok alfabe Carl 
 Faulmann'{\i}n \emph{Yaz\i\ Kitab\i}'nda g\"or\"ulebilir (T\"urkiye \.I\c s Bankas\i\ K\"ult\"ur Yay\i nlar\i, 2001).}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Kiril alfabesi, 1918'de Sovyet y\"onetiminin birka\c c harf{}i kald{\i}rmas{\i} gibi, baz{\i} de\u gi\c sikliklerin ard{\i}ndan, 33 harf{}li
 g\"un\"um\"uz Rus alfabesine d\"on\"u\c sm\"u\c st\"ur:
 \begin{center}
\selectlanguage{russian}{
A\gap B\gap V\gap G\gap D\gap E\gap \"E\gap Zh\gap Z\gap I\gap
\CYRISHRT\gap K\gap L\gap M\gap N\\O\gap P\gap R\gap S\gap T\gap
  U\gap F\gap H\gap C\gap Q\gap X\gap W\gap \CYRHRDSN\gap Y\gap
  \CYRSFTSN\gap \CYREREV\gap Yu\gap Ya}
 \end{center}
 Bu alfabe 24 harf{}li Yunan alfabesinin 19 harf{}ini, ayn{\i} s{\i}rayla ama baz{\i}lar{\i}n{\i}n formlar{\i} de\u gi\c smi\c s \c sekilde  korumaktad{\i}r. 
 \begin{compactenum}
 \item
   Yunan alfabesinin 24 harf{}i nelerdir?
   \item
   A\c sa\u g{\i}daki ki\c silerin isimleri T\"urk\c ce'de nelerdir?
   \begin{center}
     \gr{JALHS}\hfill\gr{PUJAGORAS}\hfill\gr{PLATWN}\\
     \gr{<IPPOKRATHS}\hfill\gr{>EUKLEIDHS}\hfill
     \gr{>ARQIMHDHS}
   \end{center}
 \end{compactenum}
\end{problem}


\begin{problem}
A\c sa\u g{\i}daki \"onermenin nesi hatal{\i}?
\begin{asparaenum}[1.]%[1. {\em ad{\i}m:}]
\item
Bir \"u\c cgende tabandaki bir d{\i}\c s a\c c{\i}n{\i}n a\c c{\i}ortay{\i} ile tabandaki kar\c s{\i}t i\c c
a\c c{\i}n{\i}n a\c c{\i}ortay{\i} kesi\c sirse olu\c sturduklar{\i} a\c c{\i}lar dik de\u gildir. 
\item
Bir $ABC$ \"u\c cgeni verilmi\c s olsun. 
\item
$BC$ kenar{\i} bir $D$ noktas{\i}na kadar
uzat{\i}lm{\i}\c s olsun. 
\item
$ABC$ a\c c{\i}s{\i}n{\i}n a\c c{\i}ortay{\i} $BE$ ve $ACD$ a\c c{\i}s{\i}n{\i}n a\c c{\i}ortay{\i}
$CF$, $G$ noktas{\i}nda kesi\c ssinler. 
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      \put(0.5,-0.5){\makebox(0,0){$B$}}
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      \put(13.4,-0.5){\makebox(0,0){$D$}}
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      \put(12,5.4){\makebox(0,0){$G$}}
    \end{picture}
    
  \end{figure}
\item
\.Iddia ediyorum ki $BGC$, $EGF$, $CGE$ ve $BGF$ a\c c{\i}lar{\i}n{\i}n hi\c cbiri dik
de\u gildir.
 \item 
$ACD$  a\c c{\i}s{\i} $ABC$ \"u\c cgeninin bir d{\i}\c s a\c c{\i}s{\i} oldu\u gundan i\c c ve
   kar\c s{\i}t $BAC$ a\c c{\i}s{\i}ndan b\"uy\"ukt\"ur. 
 \item 
 $ABC$ bir \"u\c cgen oldu\u gundan $ABC$ ve $BAC$ a\c c{\i}lar{\i} iki dik a\c c{\i}dan k\"u\c c\"ukt\"ur.
 \item 
 7.\ ad{\i}ma g\"ore $ABC$ ve $BAC$ a\c c{\i}lar{\i} iki dik a\c c{\i}dan k\"u\c c\"uk oldu\u gundan, $BAC$ a\c c{\i}s{\i} da iki dik a\c c{\i}dan k\"u\c c\"ukt\"ur.
 \item
 $CF$, $ACD$ a\c c{\i}s{\i}n{\i}n a\c c{\i}ortay{\i} oldu\u gu i\c cin $ACF$ ve $FCD$ a\c c{\i}lar{\i} birbirine e\c sittir.
 \item
 $CF$, $ACD$ a\c c{\i}s{\i}n{\i}n a\c c{\i}ortay{\i} oldu\u gu i\c cin $ACF$ ve $FCD$ a\c c{\i}lar{\i} birlikte $ACD$ a\c c{\i}s{\i}na e\c sittir.
 \item
 6. ve 10. ad{\i}mlara g\"ore $ACF$ ve $FCD$ a\c c{\i}lar{\i} birlikte $BAC$ a\c c{\i}s{\i}ndan b\"uy\"ukt\"ur.
 \item
 8. ve 11. ad{\i}mlara g\"ore $ACF$ ve $FCD$ a\c c{\i}lar{\i} birlikte iki dik a\c c{\i}dan k\"u\c c\"ukt\"ur.
 \item 
 9. ve 12. ad{\i}mlara g\"ore $FCD$ a\c c{\i}s{\i} bir dik a\c c{\i}dan k\"u\c c\"ukt\"ur.
 \item 
 $GCD$ a\c c{\i}s{\i} $BCG$ \"u\c cgeninin bir d{\i}\c s a\c c{\i}s{\i} oldu\u gu i\c cin $BCG$ \"u\c cgeninin i\c c ve kar\c s{\i}t $CGB$ a\c c{\i}s{\i}ndan b\"uy\"ukt\"ur.
 \item 
 13. ve 14. ad{\i}mlara g\"ore $CGB$ a\c c{\i}s{\i} bir dik a\c c{\i}dan k\"u\c c\"ukt\"ur. Dolay{\i}s{\i}yla dik de\u gildir.
 \item 
 $CGB$ ve $BGF$ a\c c{\i}lar{\i} ayn{\i} do\u gruda olduklar{\i}ndan ve 15. ad{\i}mdan $BGF$ a\c c{\i}s{\i} bir dik a\c c{\i}dan b\"uy\"ukt\"ur. Dolay{\i}s{\i}yla dik de\u gildir.
 \item 
 $FGE$ ve $CGB$ a\c c{\i}lar{\i} ters a\c c{\i} olduklar{\i} i\c cin birbirine e\c sittir. Dolay{\i}s{\i}yla 15. ad{\i}mdan $FGE$ dik de\u gildir.
 \item 
 $EGC$ ve $BGF$ a\c c{\i}lar{\i} ters a\c c{\i} olduklar{\i} i\c cin birbirine e\c sittir. Dolay{\i}s{\i}yla 16. ad{\i}mdan $EGC$ dik de\u gildir.
\end{asparaenum}
\end{problem}

\begin{problem} A\c sa\u g{\i}daki \"onermeyi tamamlay{\i}n:

\emph{\.Ilan:}
E\c skenar \"u\c cgenlerde bir kenarortay \"u\c cgenin bir kenar{\i}ndan k\"u\c c\"uk ve \"u\c cgenin bir kenar{\i}n{\i}n yar{\i}s{\i}ndan b\"uy\"ukt\"ur.

\emph{A\c c{\i}klama:} $ABC$ e\c skenar \"u\c cgeni verilmi\c s olsun. $BC$ kenar{\i}n{\i}
ikiye b\"olen $AD$ kenarortay{\i} \c cizilmi\c s olsun.

\emph{Belirtme:}  \.Iddia ediyorum ki $AD$, $AB$ kenar{\i}ndan k\"u\c c\"ukt\"ur ve
$BD$, $AD$ kenarortay{\i}ndan k\"u\c c\"ukt\"ur.

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\end{problem}


\begin{problem}
Verilmi\c s e\c skenar \"u\c cgene e\c sit olan
bir dikd\"ortgen in\c sa edin.  (Bu, \"Oklid'in \textsc i.42.\ \"onermesinin
\"ozel bir durumudur.  Bu \"onerme d\i\c s\i ndaki \"onermeleri
kullanabilirsiniz.)
\end{problem}


\begin{problem}
A\c sa\u g\i daki problemleri \c c\"oz\"un.  (Bu, \"Oklid'in
\textsc{iii}.17.\ \"onermesidir.  
  Bu \"onerme d\i\c s\i ndaki \"onermeleri
kullanabilirsiniz.) 
  \begin{compactenum}[a)]
  \item 
Verilmi\c s \c cemberdeki verilmi\c s noktadan de\u gen do\u gruyu \c
cizmek.
\item
Verilmi\c s  daire d\i\c s\i ndaki verilmi\c s noktadan de\u gen
do\u gruyu \c 
cizmek.
  \end{compactenum}
\end{problem}


\begin{problem}
Fig\"urde
\begin{compactitem}
\item 
$CBD$, merkezi $A$ olan bir \c cember;
\item 
$CAD$, merkezi $B$ olan bir \c cember;
\item
$AEB$ ile $CED$, do\u gru \c cizgiler
\end{compactitem}
olsun.  $AEC$ a\c c\i s\i n\i n dik a\c c\i\ oldu\u gunu g\"osterin.

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\uput[r](2,1){$B$}
\uput[u](1.5,1.82){$C$}
\uput[d](1.5,0.18){$D$}
\uput[ur](1.5,1){$E$}
\end{pspicture}
\end{problem}


\begin{problem}
Fig\"urde
\begin{compactitem}
\item
$AB=DE$, $BC=EF$, $AC=DF$, $BG=EH$, $AG=KH$;
\item
$BGA$ ile $EHK$ a\c c\i lar\i, dik
\end{compactitem}
olsun.  Fig\"ur\"un imk\^ans\i z oldu\u gunu g\"osterin, yani $K$ ile
$D$ noktalar\i n\i n farkl\i\ noktalar olamad\i\u g\i
n\i\ g\"osterin. 
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\psline(2,0)(2,2)
\uput[u](2,2){$A$}
\uput[d](0,0){$B$}
\uput[d](3,0){$C$}
\uput[d](2,0){$G$}
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\qquad\qquad
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\psline(1.7,0)(1.7,2)
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\uput[d](3,0){$F$}
\uput[d](1.7,0){$H$}
\end{pspicture}
\end{center}
\end{problem}



\begin{problem}
\mbox{}
\begin{compactitem}
\item
A\c c\i lar\i n\i n biri dik olan,
\item
en k\i sa kenar\i\ verilmi\c s do\u gru olan,
\item
o kenar\i\ g\"oren a\c c\i s\i\ kalan dik olmayan a\c c\i s\i n\i n yar\i s\i\ olan
\end{compactitem}
bir \"u\c cgen in\c sa edin.  (Yani en k\i sa kenar\i\ verilmi\c s bir
do\u gru olan bir `30-60-90' \"u\c cgen in\c sa edin.)  
\end{problem}


\begin{problem}
A\c sa\u g{\i}daki \"onermede hangi ad\i m, \"Oklid'in ilk 4 post\"ulat\i\ ve ilk
20 \"onermesi kullan\i larak do\u grulanamaz?
\begin{compactenum}[1.]
\item
T\"um e\c skenar \"u\c cgenlerin t\"um a\c c\i lar\i n\i n birbirine
e\c sit oldu\u gunu g\"osterece\u giz. 
\item
$ABC$ ile $DEF$, e\c skenar \"u\c cgenler olsun.
\item
$A$ a\c c\i s\i n\i n $D$ a\c c\i s\i na e\c sit oldu\u gunu g\"osterece\u giz.
\item
$AB=DE$ ise, $BC=DF$ ve $AC=DF$ de olur.
\item
O halde $A$ a\c c\i s\i\ $D$ a\c c\i s\i na e\c sit olmal\i.
\item
$AB<DE$ olsun.
\item
O zaman $AC<DF$.
\item
$DE$ kenar\i ndan $AB$ kenar\i na e\c sit olan $DG$ do\u grusu kesilsin.
\item
$DF$ kenar\i ndan $AC$ kenar\i na e\c sit olan $DH$ do\u grusu kesilsin.
\item
$G$ ile $H$ noktalar\i\ birle\c stirilsin.
\item
O zaman $DGH$ \"u\c cgeni, eskenar \"u\c cgendir.
\item
$GH=BC$.
\item
$A$ a\c c\i s\i, $D$ a\c c\i s\i na e\c sittir.
\item
$DE<AC$ ise, benzer \c sekilde $A$ a\c c\i s\i, $D$ a\c c\i s\i na e\c
  sittir.
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%      \psgrid
      \pspolygon(1,4)(3,4)(2,5.7)
      \uput[dl](1,4){$A$}
      \uput[d](3,4){$B$}
      \uput[u](2,5.7){$C$}
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    \hfill
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      \pspolygon(0,0)(3,0)(1.5,2.6)
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\hfill\mbox{}
\end{figure}
  \end{compactenum}
\end{problem}

\begin{problem}
A\c sa\u g\i daki sorular\i n \emph{biriyi} \c c\"oz\"un.
\begin{compactenum}[A.]
\item\sloppy
A\c sa\u g{\i}daki bitkilerin ve deniz hayvanlar\i n\i n isimleri T\"urk\c ce'de nedir?
\begin{center}
  \gr{AGKINARA}, \gr{ASTAKOS}, \gr{KASTANIA}\\
  \gr{MUDIA}, \gr{PRASO}, \gr{STREIDIA}
\end{center}
\item
  Yunan alfabesini s\i ras\i nda yaz\i n.
  \item
  \"Oklid'in ilk 6 \"onermesi i\c cin, kulland\i\u g\i\ \"onermeleri (ve sadece onlar\i) yaz\i n.
\end{compactenum}
\end{problem}

\begin{problem}%\sloppy
  Bir al\i\c st\i rmaya g\"ore,
verilmi\c s bir d\"ortgen bir paralelkenar ise,
o zaman k\"o\c segenleri birbirini ikiye b\"oler.
Bu \"onermenin tersini g\"osterin.
\end{problem}


\begin{problem}\sloppy
$ABC$ \"u\c cgeninde $ABC$ a\c c\i s\i\ dik olsun.
$CB$'nin, $AC$'den k\"u\c c\"uk oldu\u gunu g\"osterin.
\end{problem}



\begin{problem}\label{prob}
\emph{\gr{ABG} ve \gr{ABD} \"u\c cgenlerinin her biri,
taban\i\ \gr{AB} olan ikizkenar \"u\c cgen olsun.
O zaman \gr{GD} do\u grusu, \gr{AB} do\u grusunu keser.}
Bu teoremin
a\c sa\u g\i daki kan\i t\i n\i n
\ref 1, \ref 2, \ref 3, ve \ref 4 ad\i mlar\i n\i\ gerek\c celendirin.

  \begin{compactenum}[1.]
  \item 
De\u gilse, postulat 2 ile \gr{AB} uzat\i ls\i n,
ve \c sekildeki gibi \gr{GD},
uzat\i lm\i\c s do\u gruyu \gr E noktas\i nda kessin.
\begin{figure}[h]
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      \pspolygon(0,0)(2,2)(-3,2)
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      \psline(0,0)(0,2)
      \uput[r](2,2){\gr G}
      \uput[d](0,0){\gr A}
      \uput[dl](0,1){\gr B}
      \uput[u](0,2){\gr E}
      \uput[l](-3,2){\gr D}
    \end{pspicture}
%    \caption{\mbox{}}\label{fig}
    
  \end{figure}
\item\label{1}
$\angle\grm{ABG}=\angle\grm{BAG}$.
\item\label{2}
$\angle\grm{ABG}+\angle\grm{BAG}<\text{ 2 dik a\c c\i}$.
\item
$\angle\grm{ABG}+\angle\grm{ABG}<\text{ 2 dik a\c c\i}$.
\item
$\angle\grm{ABG}<\text{ dik a\c c\i}$.
\item
Ayn\i\ \c sekilde $\angle\grm{ABD}<\text{ dik a\c c\i}$.
\item\label{3}
$\angle\grm{BEG}+\angle\grm{BED}=\text{ 2 dik a\c c\i}$.
\item
\gr{BEG} ve \gr{BED} a\c c\i lar\i n\i n biri,
ya dik ya da dik a\c c\i dan k\"u\c c\"ukt\"ur.
\item
$\angle\grm{BEG}\geqslant\text{ dik a\c c\i}$ olsun.
\item\label{4}
$\angle\grm{ABG}>\angle\grm{BEG}$.
\item
Yani dik a\c c\i dan k\"u\c c\"uk olan a\c c\i,
 dik a\c c\i dan k\"u\c c\"uk olmayan a\c c\i dan b\"uy\"ukt\"ur.
\item
Ki bu imk\^ans\i zd\i r.
  \end{compactenum}

\end{problem}



\begin{problem}
\emph{Verilmi\c s do\u gruyu ikiye b\"olmek.}
Bu problemin
a\c sa\u g\i daki \c c\"oz\"um\"un\"un yanl\i\c s\i n\i\ 
veya yanl\i\c slar\i n\i\ d\"uzeltin.

  \begin{compactenum}[1.]
  \item 
Verilmi\c s do\u gru, $AB$ olsun.
\item
\"Onerme 1'i kullanarak
$ABC$ ve $ABD$ e\c skenar \"u\c cgen olsun.
\begin{figure}[h]
  \centering
    \psset{unit=1cm}
    \begin{pspicture}(-2.23,-1.5)(2.23,1.5)
      %\psgrid
      \pspolygon(0,-1)(-1.73,0)(0,1)(1.73,0)
      \psline(0,-1)(0,1)
      \psline(1.73,0)(-1.73,0)
      \uput[d](0,-1){$A$}
      \uput[u](0,1){$B$}
      \uput[r](1.73,0){$C$}
      \uput[l](-1.73,0){$D$}
      \uput[ur](0,0){$E$}
    \end{pspicture}
%    \caption{}
    
  \end{figure}
\item
  \ref{prob} numaral\i\ soruya g\"ore
  $CD$ ve $AB$ do\u grular\i\ bir $E$ noktas\i nda kesi\c sir.
\item
\"Onerme 5'e g\"ore $\angle ACD=\angle BCD$.
\item
\"Onerme 6'ya g\"ore $AE=EB$.
  \end{compactenum}
\end{problem}



\begin{problem}
  $\grm{GB}\neq\grm{DB}$ ve $\grm{AG}=\grm{AD}$ olacak \c sekilde 
iki $\grm{ABG}$ ve $\grm{ABD}$ \"u\c cgeni in\c sa ediniz.  
\.In\c san\i z\i n do\u gru oldu\u gunu g\"osterin.
Yaln\i z \"Oklid'in tan\i m (s\i n\i r), postulat, ortak kavram, 
ve \"onermelerini kullan\i n.
\end{problem}


\begin{problem}
A\c sa\u g\i daki \"onerme do\u gru mu, yanl\i\c s m\i?
A\c c\i klay\i n.
%\begin{minipage}{0.65\textwidth}
\begin{compactenum}[1.]
\item
D\i\c sb\"ukey bir d\"ortgenin k\"o\c segeni \c cizilirse,
yan\i ndaki a\c c\i,
g\"ord\"u\u g\"u a\c c\i dan k\"u\c c\"ukt\"ur.
\item
D\"ortgen $ABCD$ olsun, ve k\"o\c segeni $BD$ olsun.
\begin{figure}[h]
  \centering
  \begin{pspicture}(-0.5,-0.2)(2.5,2.2)
%    \psgrid
\psline(0,0)(2,0)(2,2)(0,2)(0,0)(2,2)
\psline(2,0)(0,2)
\uput[l](0,0){$A$}
\uput[r](2,0){$B$}
\uput[r](2,2){$C$}
\uput[l](0,2){$D$}
\end{pspicture}

\end{figure}
\item
Diyorum ki $BDC$ a\c c\i s\i,
$BCD$ a\c c\i s\i ndan k\"u\c c\"ukt\"ur.
\item
Zira
$AC$ k\"o\c segeni \c cizilsin.
\item
$BAC$ a\c c\i s\i, $BAD$ a\c c\i s\i ndan k\"u\c c\"ukt\"ur.
\item\label{hata}
Dolay\i s\i yla $BC<BD$.
\item
Bu durumda $BDC$ a\c c\i s\i,
$BCD$ a\c c\i s\i ndan k\"u\c c\"ukt\"ur.
\end{compactenum}
\end{problem}


\begin{problem}
\.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 oldu\u gunu g\"osterin.
\end{problem}


\begin{problem}
\.Iki dik \"u\c cgende
dik a\c c\i y\i\ g\"oren kenar
dik a\c c\i y\i\ g\"oren kenara e\c sitse ve
di\u ger bir kenar
di\u ger bir kenara e\c sitse,
o zaman kalan kenar\i n
kalan kenara e\c sit olaca\u g\i n\i\ kan\i tlay\i n.
\end{problem}


\begin{problem}
Bir \"u\c cgenin k\"o\c selerinden ge\c cen bir \c cember \c cizin.
(Tabii ki in\c san\i z\i n neden do\u gru oldu\u gunu a\c c\i klamal\i s\i n\i z.)
\end{problem}


\begin{problem}
$AB$ ve $AC$ do\u grular\i n\i n kesi\c simi ile olu\c san bir 
$BAC$ a\c c\i s\i\ ve bir $D$ noktas\i\ verilmi\c s olsun. 
$D$ noktas\i ndan ge\c cen ve
($A$'n\i n taraf\i nda) 
$AB$, $AC$ do\u grular\i ndan 
e\c sit par\c calar ay\i ran bir do\u gru 
\c cizilebilece\u gini g\"osteriniz.  \emph{\.Ipucu}: 
$BAC$ a\c c\i s\i n\i\ ikiye b\"olen do\u gruyu \c cizerek ba\c slayabilirsiniz.
\begin{figure}[h]
  \psset{unit=6mm}
  \begin{center}
    \begin{pspicture}(-0.6,-0.6)(4.6,4.6)
      %\psgrid
      \pstGeonode(4,2)D
      \psset{PointSymbol=none}
      \pstGeonode[PosAngle={-90,135}](1,0)A(0,4)B
      \pstRotation[RotAngle=-41,PosAngle=30] AB[C]
      \ncline AB\ncline AC
    \end{pspicture}
  \end{center}
  
\end{figure}
\end{problem}


\begin{problem}
Bir $ABC$ \"u\c cgeninin
$AB$ taban\i na paralel olan
ve di\u ger kenarlar\i\ kesen bir do\u gru $DE$ olsun,
ve $AB$'nin orta noktas\i\ $F$ olsun.
$CF$nin ve $DE$'nin kesi\c sim noktas\i\ $G$ olsun.
$G$'nin $DE$'nin orta noktas\i\ oldu\u gunu kan\i tlay\i n.
\"Onerme 36'n\i n tersi kabul edilebilir.
\emph{\.Ipucu}: $AH\parallel FC\parallel BK$ ve $HK\parallel AB$ olsun;
$DE$, $L$ ve $M$'ye uzat\i ls\i n;
ve s\i ras\i yla $D$ ve $E$'den ge\c cen,
$CF$'ye paralel olan $NP$ ve $QR$ \c cizilsin.
$NF$ ve $FQ$ paralelkenarlar\i n\i n e\c sit oldu\u gunu g\"osterin.
\begin{figure}[h]
  \begin{center}
    \psset{unit=6mm}
    \begin{pspicture}(-0.6,-0.6)(9.6,6.6)%\psgrid
      \psset{PointSymbol=none}
      \pstTriangle[PosAngleA=-90,PosAngleB=-90,PosAngleC=90] (0,0)A(8,0)B(5,6)C
      \pstMiddleAB BAF
      \ncline CF
      \pstTranslation[PosAngle={90,90}] FC{A,B}[H,K]
      \pstHomO[HomCoef=0.7,PosAngle={-45,-135,90,90}] C{A,B,H,K}[D,E,N,Q]
      \pstInterLL[PosAngle=135] DECFG
      \ncline DE
      \psset{linestyle=dashed}
      \ncline AH\ncline HK\ncline KB
      \pstInterLL[PosAngle=180] AHDEL
      \pstInterLL BKDEM
      \ncline LD\ncline EM
      \psset{PosAngle=-90}
      \pstInterLL NDABP
      \pstInterLL QEABR
      \ncline NP\ncline QR
    \end{pspicture}
  \end{center}

  
\end{figure}
\end{problem}




\end{document}
