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\usepackage[turkish]{babel}
\usepackage{hfoldsty}
\usepackage[neverdecrease]{paralist}
\usepackage{pstricks,pst-eucl,wrapfig}
\usepackage{amsmath,amsthm,amssymb,upgreek,bm,mathrsfs}

\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem*{solution}{\c C\"oz\"um}

\usepackage{verbatim}
%\let\solution=\comment
%\let\endsolution=\endcomment

\newcommand{\lto}{\rightarrow}
\newcommand{\liff}{\leftrightarrow}
\pagestyle{empty}
\begin{document}

\title{\"Oklid Geometrisine Giri\c s}
\subtitle{Final S\i nav\i\ \emph{\c C\"oz\"umleri}}
\date{5 Ocak 2017}
\author{K\i van\c c Ersoy \and David Pierce \and G\"ulay Telsiz \and \.Ipek Tuvay}
\maketitle
\thispagestyle{empty}

Buradaki d\"ort problemden sadece \"u\c c tanesini \c c\"oz\"un.

\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{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{problem}

\begin{solution}\mbox{}\\
\begin{pspicture}(-0.6,0)(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
\pstBissectBAC CABE
\pstProjection[PosAngle=135] AED[F]
\pstRightAngle[RightAngleSize=0.21] DFE
\pstInterLL[PosAngle=180] DFABG
\pstInterLL[PosAngle=-45] DFACH
\ncline DG
\end{pspicture}
\hfill
\begin{minipage}[b]{0.65\textwidth}
\begin{compactenum}
\item
$BAC$ a\c c\i s\i\ $AE$ do\u grusu ile ikiye b\"ol\"uns\"un [9].
\item
$AE$'ye dik olan $DF$ indirilsin [12].
\item
Bu $DF$ do\u grusu, $AB$'yi $G$'de ve $AC$'yi $H$'de kessin.
\item
$AFG$ ve $AFH$ a\c c\i lar\i\ dik oldu\u gundan e\c sittir.
\item
$FAG$ ve $HAG$ a\c c\i lar\i\ da e\c sit oldu\u gundan,
ve $AF$ ortak oldu\u gundan,
$AFG$ ve $AFH$ \"u\c cgenlerinde $AG=AH$ [26].
\end{compactenum}
\end{minipage}
\end{solution}

\newpage

\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{center}
\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{problem}

\begin{solution}\mbox{}
[E\u ger bir $XY$ do\u grusu 
bir paralelkenar\i n \c cizilmemi\c s k\"o\c segeniyse,
o zaman $XY$ bu paralelkenar\i n ad\i d\i r.]

\begin{center}
\begin{minipage}{0.6\textwidth}
\begin{compactenum}
\item
$HF=FK$ ve $LF=FM$\hfill[36]
\item
$HG=GK$\hfill[O.K. 3]
\item
$HD=DF$ ve $FE=EK$\hfill[43]
\item
$HG=NF$ ve $FQ=GK$\hfill[O.K. 2]
\item
$NF=FQ$\hfill[O.K. 1]
\item
$PF=FR$\hfill[36'n\i n tersi]
\item
$PF=DG$ ve $FR=GE$\hfill[34]
\item
$DG=GE$\hfill[O.K. 1]
\end{compactenum}
\end{minipage}
\end{center}
\end{solution}

\newpage

\begin{problem}
\begin{compactenum}[(a)]
\item
A\c sa\u g\i daki form\"ullerin ikisinin her biri,
di\u gerinin de\u gillemesine denktir.
Hangi form\"ul, hangi form\"ul\"un de\u gillemesine denktir?
\begin{align*}
	\lnot Q&\land(R\lto\lnot P),&
	\lnot P\lor\lnot R&\lto Q,&
	P\lor Q&\lto\lnot(P\land Q).
\end{align*}
\item
Bi\c cimsel kan\i t ile
\begin{equation*}
R\liff P\lor S,\ R\lto Q\models P\lto Q
\end{equation*}
gerektirmesini g\"osterin.
\end{compactenum}
\end{problem}

\begin{solution}
  (a)
  Verilen form\"ullerin do\u gruluk tablolar\i
\begin{equation*}
\begin{array}{*2c|c|*4c||*4c|c|c||*3c|c|*4c}
\lnot&\multicolumn1cQ&\multicolumn1c{\land}&(R&\lto&\lnot&P)& 
\lnot&(P&\land&\multicolumn1c{R)}&\multicolumn1c{\lto}&Q& 
P&\lor&\multicolumn1cQ&\multicolumn1c{\lto}&\lnot&(P&\land&Q)\\\hline
   1 &0&  1  & 0&  1 &  1  &0 &   1  & 0&  0  &0 & 0  &0& 0&  0 &0&  1 &  1  & 0&  0  &0 \\
   1 &0&  1  & 0&  1 &  0  &1 &   1  & 1&  0  &0 & 0  &0& 1&  1 &0&  1 &  1  & 1&  0  &0 \\
   0 &1&  0  & 0&  1 &  1  &0 &   1  & 0&  0  &0 & 1  &1& 0&  1 &1&  1 &  1  & 0&  0  &1 \\
   0 &1&  0  & 0&  1 &  0  &1 &   1  & 1&  0  &0 & 1  &1& 1&  1 &1&  0 &  0  & 1&  1  &1 \\
   1 &0&  1  & 1&  1 &  1  &0 &   1  & 0&  0  &1 & 0  &0& 0&  0 &0&  1 &  1  & 0&  0  &0 \\
   1 &0&  0  & 1&  0 &  0  &1 &   0  & 1&  1  &1 & 1  &0& 1&  1 &0&  1 &  1  & 1&  0  &0 \\
   0 &1&  0  & 1&  1 &  1  &0 &   1  & 0&  0  &1 & 1  &1& 0&  1 &1&  1 &  1  & 0&  0  &1 \\
   0 &1&  0  & 1&  0 &  0  &1 &   0  & 1&  1  &1 & 1  &1& 1&  1 &1&  0 &  0  & 1&  1  &1\\
	\cline{3-3}\cline{12-12}\cline{17-17}
\end{array}
\end{equation*}
oldu\u gundan
\begin{equation*}
 \lnot Q\land(R\lto\lnot P)\sim\lnot(\lnot P\lor\lnot R\lto Q). 
\end{equation*}
Ayr\i ca
\begin{multline*}
  \lnot(\lnot P\lor\lnot R\lto Q)
  \sim\lnot\bigl(\lnot(\lnot P\lor\lnot R)\lor Q\bigr)\\
  \sim(\lnot P\lor\lnot R)\land\lnot Q
  \sim(R\lto\lnot P)\land\lnot Q
  \sim\lnot Q\land(R\lto\lnot P).
\end{multline*}

(b)
\begin{tabular}[t]{rcl}
1.&$R\liff P\lor S$&hipotez\\
2.&$P\lor S\lto R$&[basitle\c stirme]\\
3.&$\lnot(P\lor S)\lor R$&\\
4.&$(\lnot P\land\lnot S)\lor R$&\\
5.&$(\lnot P\lor R)\land(\lnot S\lor R)$&\\
6.&$\lnot P\lor R$&basitle\c stirme\\
7.&$P\lto R$&\\
8.&$R\lto Q$&hipotez\\
9.&$P\lto Q$&hipotetik tas\i m
\end{tabular}
\end{solution}


\newpage

\begin{problem}
\begin{compactenum}[(a)]
\item
A\c sa\u g\i daki verilen do\u gruluk tablosunu doldurun.
\item
En sevdi\u giniz y\"ontemi kullanarak $Q\lto R$ form\"ul\"un\"un
$\lnot P\lto Q\lto R\land 1$ form\"ul\"un\"u gerektirdi\u gini g\"osterin.
\item
Zaten yapmad\i ysan\i z,
$Q\lto R\models\lnot P\lto Q\lto R\land 1$ gerektirmesini bi\c cimsel kan\i t ile g\"osterin.
\end{compactenum}
\begin{equation*}
\renewcommand{\arraystretch}{1.5}
\begin{array}{*8c}
\lnot&P&\lto&Q&\lto&R&\land&1\\\hline
     &0&    &0&    &0&     & \\
     &1&    &0&    &0&     & \\\hline
     &0&    &1&    &0&     & \\
     &1&    &1&    &0&     & \\\hline
     &0&    &0&    &1&     & \\
     &1&    &0&    &1&     & \\\hline
     &0&    &1&    &1&     & \\
     &1&    &1&    &1&     & \\\hline
\end{array}
\end{equation*}
\end{problem}

\begin{solution}\mbox{}\\
(a)
\begin{math}
\begin{array}[t]{cc|c|*5c}
\lnot&\multicolumn1cP&\multicolumn1c{\lto}&Q&\lto&R&\land&1\\\hline
   1 &0&  1 &0& 1  &0&  0  &1\\
   0 &1&  1 &0& 1  &0&  0  &1\\
   1 &0&  0 &1& 0  &0&  0  &1\\
   0 &1&  1 &1& 0  &0&  0  &1\\
   1 &0&  1 &0& 1  &1&  1  &1\\
   0 &1&  1 &0& 1  &1&  1  &1\\
   1 &0&  1 &1& 1  &1&  1  &1\\
   0 &1&  1 &1& 1  &1&  1  &1\\\cline{3-3}
\end{array}
\end{math}
\hfill
(b)
\begin{math}
\begin{array}[t]{c|c|c}
\multicolumn1cQ&\multicolumn1c{\lto}&R\\\hline
0&1&0\\
0&1&0\\
1&0&0\\
1&0&0\\
0&1&1\\
0&1&1\\
1&1&1\\
1&1&1\\\cline{2-2}
\end{array}
\end{math}\hfill
\begin{minipage}[t]{0.35\textwidth}
($Q\lto R$ form\"ul\"un\"un do\u gru oldu\u gu her sat\i rda
$\lnot P\lto Q\lto R\land 1$ de do\u grudur)  
\end{minipage}

(c)
\begin{math}
  \begin{gathered}[t]
    Q\lto R\\
    Q\lto R\land1\\
    P\lor(Q\lto R\land 1)\\
    \lnot P\lto Q\lto R\land 1
  \end{gathered}
\end{math}
\end{solution}

\end{document}
