\documentclass[% version=last,% a4paper,% 12pt,% headings=small,% % twoside,% % open=any,% parskip=half,% this option takes 2.5% more space than parskip % draft=true,% %DIV=12,% headinclude=false,% pagesize]% {scrartcl} %\documentclass[a4paper,12pt]{article} \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}