<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\NeedsTeXFormat{LaTeX2e}% lualatex

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\RequirePackage{mathtools,amssymb,ifthen,xparse,tikz,graphics}%
\usepackage[b]{esvect}%
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\def\ç›¸ä¼¼sizeratio#1{\Mulself\sz@s{#1}\Mulself\sz@r{#1}\ignorespaces}%
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\newcommand{\Tzettaiti}[1]{\left|\,#1\,\right|}%
\def\shikimaru#1{\text{\quad$\cdots\cdots$\,\ajMaru{#1}}}
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\long\def\@colonfor#1:=#2\do#3{% % \@for ã®åŒºåˆ‡ã‚Šè¨˜å·ã‚’ : ã«å¤‰ãˆãŸãƒãƒ¼ã‚¸ãƒ§ãƒ³
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\[n\times\Ttyuukakko{n-1}\times\Ttyuukakko{n-2}\times\cdots\times\Ttyuukakko{n-r+1}=\bunsuu{n\éšŽä¹—}{\Ttyuukakko{n-r}\éšŽä¹—}\]%
  
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\[{}_{n}\text{C}_{r}=\bunsuu{n\éšŽä¹—}{r\éšŽä¹—\Ttyuukakko{n-r}\éšŽä¹—}\]%

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\[\bunsuu{{}_{n} P_{r}}{r\éšŽä¹—}=\bunsuu{n\éšŽä¹—}{r\éšŽä¹—\Ttyuukakko{n-r}\éšŽä¹—}\]%

        ã“ã“ã§ï¼Œ$\bunsuu{n\éšŽä¹—}{r\éšŽä¹—\Ttyuukakko{n-r}\éšŽä¹—}$ã‚’${}_{n}\text{C}_{r}$ã¨è¡¨ã™ã€‚        \è¨¼æ˜Žçµ‚äº†%
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\[\bunsuu{n\éšŽä¹—}{p\éšŽä¹— q\éšŽä¹— r\éšŽä¹—}\text{\ ï¼ˆãŸã&nbsp;ã—ï¼Œ$p+q+r=n$ï¼‰}\]%

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      {%
        äº‹è±¡$A$ã¨$B$ãŒç‹¬ç«‹ã®ã¨ãï¼Œäº‹è±¡$A$ãŒèµ·ã“ã‚Šã‹ã¤äº‹è±¡$B$ãŒèµ·ã“ã‚‹ç¢ºçŽ‡$p$ã¯ï¼Œ%
\[p=P\Ttyuukakko{A}P\Ttyuukakko{B}\]%

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      {%
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      {%
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          \draw(-0.4,2.8)--(0,0);%
          \draw(0,0)node[below]{I};%
          \draw(-1,2)--(0,0);%
          \draw(0.666666,2)--(0,0);%
          \draw(0,2)node[above]{D};%
          \draw(-1.6,1.2)node[left]{E};%
          \draw(1.2,1.6)node[right]{F};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        å›³ã«ãŠã„ã¦IãŒå‚å¿ƒ%
      }%
      {\relax}% 
    \ifthenelse{\equal{#1}{ãƒã‚§ãƒã®å®šç†}}%
      {%
        \begin{tikzpicture}
          \draw(2,3)--(0,0)--(3,0)--cycle;%
          \draw(2,3)node[above]{A};%
          \draw(2,3)--(1.5,0);%
          \draw(0,0)node[below]{B};%
          \draw(0,0)--(2.75,0.75);%
          \draw(3,0)node[below]{C};%
          \draw(3,0)--(0.5,0.75);%
          \draw(1.5,0)node[below]{P};%
          \draw(2.75,0.75)node[right]{Q};%
          \draw(0.5,0.75)node[left]{R};%
          \draw(1.62,0.8)node[right]{O};%
        \end{tikzpicture}
        \ç©ºè¡Œ%
        $\bunsuu{\text{BP}}{\text{PC}}\cdot\bunsuu{\text{CQ}}{\text{QA}}\cdot\bunsuu{\text{AR}}{\text{RB}}=1$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{ãƒã‚§ãƒã®å®šç†ã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \begin{tikzpicture}%
          \draw(2,3)--(0,0)--(3,0)--cycle;%
          \draw(2,3)node[above]{A};%
          \draw(2,3)--(1.5,0);%
          \draw(0,0)node[below]{B};%
          \draw(0,0)--(2.75,0.75);%
          \draw(3,0)node[below]{C};%
          \draw(3,0)--(0.5,0.75);%
          \draw(1.5,0)node[below]{P};%
          \draw(2.75,0.75)node[right]{Q};%
          \draw(0.5,0.75)node[left]{R};%
          \draw(1.62,0.8)node[right]{O};%   
        \end{tikzpicture}%  
        \ç©ºè¡Œ%
        å›³ã«ãŠã„ã¦ä¸‰è§’å½¢ã®é¢ç©æ¯”ã‚’è€ƒãˆã‚‹ã¨ï¼Œ%
\[\bigtriangleup{ABO}:\bigtriangleup{ACO}=\text{BP}:\text{CP}\]%
\[\Leftrightarrow\bunsuu{\bigtriangleup{ABO}}{\bigtriangleup{ACO}}=\bunsuu{\text{BP}}{\text{PC}}\]%

        åŒæ§˜ã«ã—ã¦ï¼Œ$\bunsuu{\bigtriangleup{BCO}}{\bigtriangleup{BAO}}=\bunsuu{\text{CQ}}{\text{QA}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{\bigtriangleup{CAO}}{\bigtriangleup{CBO}}=\bunsuu{\text{AR}}{\text{RB}}$%

        ã“ã“ã§ï¼Œ%
\[\bunsuu{\bigtriangleup{ABO}}{\bigtriangleup{ACO}}\cdot\bunsuu{\bigtriangleup{BCO}}{\bigtriangleup{BAO}}\cdot\bunsuu{\bigtriangleup{CAO}}{\bigtriangleup{CBO}}=1\]%
\[\Leftrightarrow\bunsuu{\text{BP}}{\text{PC}}\cdot\bunsuu{\text{CQ}}{\text{QA}}\cdot\bunsuu{\text{AR}}{\text{RB}}=1\]%
           \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}% 
    \ifthenelse{\equal{#1}{ãƒ¡ãƒãƒ©ã‚¦ã‚¹ã®å®šç†}}%
      {%
        \begin{tikzpicture}%
          \draw(2,3)--(0,0)--(3,0)--cycle;%
          \draw(2,3)node[above]{A};%
          \draw(0,0)node[below]{B};%
          \draw(3,0)node[right]{C};%
          \draw(3,0)--(3.11111,-0.33333);%
          \draw(2.25,0)node[below]{P};%
          \draw(3.11111,-0.33333)node[below]{Q};%
          \draw(3.11111,-0.33333)--(0.5,0.75);%
          \draw(0.5,0.75)node[left]{R};%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\bunsuu{\text{BP}}{\text{PC}}\cdot\bunsuu{\text{CQ}}{\text{QA}}\cdot\bunsuu{\text{AR}}{\text{RB}}=1$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{ãƒ¡ãƒãƒ©ã‚¦ã‚¹ã®å®šç†ã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \begin{tikzpicture}%
          \draw(2,3)--(0,0)--(3,0)--cycle;%
          \draw(2,3)node[above]{A};%
          \draw(0,0)node[below]{B};%
          \draw(3,0)node[right]{C};%
          \draw(3,0)--(3.11111,-0.33333);%
          \draw(2.25,0)node[below]{P};%
          \draw(3.11111,-0.33333)node[below]{Q};%
          \draw(3.11111,-0.33333)--(0.5,0.75);%
          \draw(0.5,0.75)node[left]{R};%
          \draw(0.65,0.975)node[above left]{S};%
          \draw(0.65,0.975)--(3,0);%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\text{SC}\å¹³è¡Œ\text{RP}$ã‚ˆã‚Šï¼Œ%
\[\text{RA}:\text{SR}=\text{QA}:\text{CQ}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\text{BR}:\text{RS}=\text{BP}:\text{PC}\]%
\[\Leftrightarrow\bunsuu{\text{CQ}}{\text{QA}}=\bunsuu{\text{SR}}{\text{AR}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{\text{BP}}{\text{PC}}=\bunsuu{\text{BR}}{\text{RS}}\]%
        \[\bunsuu{\text{BP}}{\text{PC}}\cdot\bunsuu{\text{CQ}}{\text{QA}}\cdot\bunsuu{\text{AR}}{\text{RB}}=\bunsuu{\text{BR}}{\text{RS}}\cdot\bunsuu{\text{SR}}{\text{AR}}
        \cdot\bunsuu{\text{AR}}{\text{RB}}=1\]%
        \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{å††å‘¨è§’ã®å®šç†}}%
      {%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(1.2,1.6)--cycle;%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(-2,0)--cycle;%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.2,-1.6)node[right]{B};%
          \draw(1.2,1.6)node[above]{P};%
          \draw(-2,0)node[left]{Q};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\angle\text{APB}=\angle\text{AQB}$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{å††å‘¨è§’ã®å®šç†ã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \begin{tikzpicture}%%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(1.2,1.6)--cycle;%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(0,0)--cycle;%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.2,-1.6)node[right]{B};%
          \draw(1.2,1.6)node[above]{P};%
          \draw(0,0)node[right]{O};%
          \draw[dashed](-1.2,-1.6)--(1.2,1.6);% 
          \draw(-0.6,-1.4)node[above]{D};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\triangle{\text{AOP}}$\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹$\triangle{\text{BOP}}$ã¯äºŒç­‰è¾ºä¸‰è§’å½¢ãªã®ã§ï¼Œ%
\[\angle\text{APO}=\angle\text{OAP}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\angle\text{BPO}=\angle\text{OBP}\]%

        å¤–è§’å®šç†ã‚ˆã‚Šï¼Œ%
\[\angle\text{AOD}=2\angle\text{APO}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\angle\text{BOD}=2\angle\text{BPO}\]%
\[\Leftrightarrow\angle\text{AOB}=2\angle\text{APB}\]%

        \ç©ºè¡Œ%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.6,-1.2)--(1.6,1.2)--cycle;%
          \draw(-1.6,-1.2)--(1.6,-1.2)--(0,0)--cycle;%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.6,-1.2)node[right]{B};%
          \draw(1.6,1.2)node[above]{P};%
          \draw(0,0)node[above]{O};%    
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\triangle{\text{OPB}}$ã¯äºŒç­‰è¾ºä¸‰è§’å½¢ãªã®ã§ï¼Œ%
\[\angle\text{OPB}=\angle\text{OBP}\]%

        å¤–è§’å®šç†ã‚ˆã‚Š%
\[\angle\text{AOB}=2\angle\text{OPB}\]%

        \ç©ºè¡Œ%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(0,0)--cycle;%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(-2,0)--cycle;% 
          \draw[dashed](-2,0)--(2,0);%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.2,-1.6)node[right]{B};%
          \draw(0,0)node[above]{O};%
          \draw(2,0)node[right]{D};%  
          \draw(-2,0)node[left]{Q};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\triangle{\text{QOA}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\triangle{\text{OQB}}$ã¯äºŒç­‰è¾ºä¸‰è§’å½¢ãªã®ã§ï¼Œ%
\[\angle\text{OQA}=\angle\text{OAQ}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\angle\text{OQB}=\angle\text{OBQ}\]%

        å¤–è§’å®šç†ã‚ˆã‚Šï¼Œ%
\[\angle\text{OQA}+\angle\text{OAQ}=\angle\text{DOA}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\angle\text{OQB}+\angle\text{OBQ}=\angle\text{DOB}\]%
\[\Leftrightarrow\angle\text{DOA}-\angle\text{DOB}=2\Ttyuukakko{\angle\text{OQA}-\angle\text{BQO}}\]%
\[\Leftrightarrow\angle\text{AOB}=2\angle\text{AQB}\]%

        å¾“ã£ã¦ï¼Œå††ã«å†…æŽ¥ã™ã‚‹ä¸‰è§’å½¢ã«ã¤ã„ã¦ï¼Œå††å‘¨è§’ã®$2$å€ãŒä¸­å¿ƒè§’ã§ã‚ã‚‹ã€‚%
        \ç©ºè¡Œ~\ç©ºè¡Œ%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(1.2,1.6)--cycle;%
          \draw(-1.6,-1.2)--(1.2,-1.6)--(-2,0)--cycle;%
          \draw(0,0)--(-1.6,-1.2)--(1.2,-1.6)--cycle;%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.2,-1.6)node[right]{B};%
          \draw(1.2,1.6)node[above]{P};%
          \draw(-2,0)node[left]{Q};%
          \draw(0,0)node[above]{O};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        ä»¥ä¸Šã‚ˆã‚Šï¼Œä»¥ä¸‹ãŒæˆç«‹ã€‚        \[\angle\text{APB}=2\angle\text{AOB},\angle\text{AQB}=2\angle\text{AOB}\]%
\[\Leftrightarrow\angle\text{AQB}=\angle\text{APB}\]
        \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}% 
      \ifthenelse{\equal{#1}{å†…æŽ¥å››è§’å½¢ã®å®šç†}}%
      {%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.6,-1.2)--(1.2,1.6)--(0,2);%
          \draw(-1.6,-1.2)--(0,2);%
          \draw(-1.6,-1.2)--(3,-1.2);%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.6,-1.2)node[below]{B};%
          \draw(1.2,1.6)node[above]{C};%
          \draw(0,2)node[above]{D};%
          \draw(3,-1.2)node[below]{T};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\angle\text{ADC}=\angle\text{CBT}$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{å†…æŽ¥å››è§’å½¢ã®å®šç†ã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \begin{tikzpicture}%
          \draw(-1.6,-1.2)--(1.6,-1.2)--(1.2,1.6)--(0,2);%
          \draw(-1.6,-1.2)--(0,2);%
          \draw(-1.6,-1.2)--(3,-1.2);%
          \draw(-1.6,-1.2)node[left]{A};%
          \draw(1.6,-1.2)node[below]{B};%
          \draw(1.2,1.6)node[above]{C};%
          \draw(0,2)node[above]{D};%
          \draw(3,-1.2)node[below]{T};%
          \draw(0,0)circle[radius=2];% 
          \draw(-1.6,-1.2)--(0,0);%
          \draw(1.2,1.6)--(0,0);%
          \draw(0,0)coordinate(O);%
        \fill[black](O)circle(0.03);%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
\[\angle\text{AOC}=2\angle\text{ABC}\]%
        \[\angle\text{AOC}=2\angle\text{ADC}\]%

        ã“ã“ã§ï¼Œ$\angle\text{ABC}+\angle\text{ADC}=180^\circ$%
\[\Leftrightarrow\angle\text{AOC}+\angle\text{AOC}=180^\circ\]%
        \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{æŽ¥å¼¦å®šç†}}%
      {%
        \begin{tikzpicture}%
          \draw(0,-2)--(2,0)--(-1.2,1.6)--cycle;%
          \draw(0,-2)node[below]{A};%
          \draw(2,0)node[right]{B};%
          \draw(-1.2,1.6)node[above]{C};%
          \draw(3,-2)--(-3,-2);%
          \draw(3,-2)node[below]{T};%
          \draw(-3,-2)node[below]{S};%
          \draw(0,0)circle[radius=2];%
        \end{tikzpicture}
        \ç©ºè¡Œ%
        $\angle\text{BAT}=\angle\text{ACB}$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{æŽ¥å¼¦å®šç†ã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \vspace{-1\zw}%
        \begin{enumerate}%
          \item é‹­è§’ã®ã¨ã%  
                \ç©ºè¡Œ% 
                \begin{tikzpicture}%
                  \draw(0,-2)--(2,0)--(-1.2,1.6)--cycle;%
                  \draw[dashed](0,-2)--(2,0)--(0,2)--cycle;%
                  \draw(0,-2)node[below]{A};%
                  \draw(2,0)node[right]{B};%
                  \draw(-1.2,1.6)node[above]{C};%
                  \draw(3,-2)--(-3,-2);%
                  \draw(3,-2)node[below]{T};%   
                  \draw(-3,-2)node[below]{S};%
                  \draw(0,2)node[above]{E};%
                  \draw(0,0)circle[radius=2];%
                  \draw(0,2)coordinate(E);%
                  \draw(2,0)coordinate(B);%
                  \draw(0,-2)coordinate(A);%
                  \draw pic[draw,black,thin,angle radius=0.3cm] {right angle=E--B--A};%
                \end{tikzpicture}%
                \ç©ºè¡Œ%
                $\triangle{\text{ACB}}$ã¨$\triangle{\text{ABE}}$ã«ã¤ã„ã¦å††å‘¨è§’ã®å®šç†ã‚ˆã‚Šï¼Œ%
                \[\angle\text{ACB}=\angle\text{AEB}\]%

                ã“ã“ã§ï¼Œ$\triangle{\text{ABE}}$ã«ã¤ã„ã¦%
                \[\angle\text{BEA}+\angle\text{BAE}=90^\circ\]%
  
                ã¾ãŸï¼Œ$\text{AT}$ãŒå††ã®æŽ¥ç·šãªã®ã§$\angle\text{BAE}+\angle\text{BAT}=90^\circ$ã‹ã‚‰ï¼Œ%
                \[\angle\text{BAT}=\angle\text{AEB}\]%
                \[\Leftrightarrow\angle\text{ACB}=\angle\text{BAT}\]%

                \ç©ºè¡Œ%
          \item ç›´è§’ã®ã¨ã%
                \ç©ºè¡Œ
                \begin{tikzpicture}%
                  \draw(0,-2)--(2,0)--(0,2)--cycle;%
                  \draw(0,-2)node[below]{A};%
                  \draw(2,0)node[right]{B};%
                  \draw(3,-2)--(-3,-2);%
                  \draw(3,-2)node[below]{T};%
                  \draw(-3,-2)node[below]{S};%
                  \draw(0,2)node[above]{E};%
                  \draw(0,0)circle[radius=2];%
                  \draw(0,2)coordinate(E);%
                  \draw(2,0)coordinate(B);%
                  \draw(0,-2)coordinate(A);%
                  \draw pic[draw,black,thin,angle radius=0.3cm] {right angle=E--B--A};%
                \end{tikzpicture}%
                \ç©ºè¡Œ%
                ATãŒå††ã®æŽ¥ç·šãªã®ã§ï¼Œ%
                \[\angle\text{EAS}=90^\circ\]%
                \[\Leftrightarrow\angle\text{EBA}=\angle\text{EAS}\]%

                \ç©ºè¡Œ%
          \item éˆè§’ã®ã¨ã%
                \ç©ºè¡Œ%
                \begin{tikzpicture}%
                  \draw(0,-2)--(-2,0)--(-1.2,1.6)--cycle;
                  \draw(0,-2)node[below]{A};%
                  \draw(-2,0)node[left]{B};%
                  \draw(-1.2,1.6)node[above]{C};%
                  \draw(3,-2)--(-3,-2);%
                  \draw(3,-2)node[below]{T};%
                  \draw(-3,-2)node[below]{S};%
                  \draw(0,0)circle[radius=2];%
                \end{tikzpicture}%
                \ç©ºè¡Œ%
                é‹­è§’ã®ã¨ãã®æŽ¥å¼¦å®šç†ã‚ˆã‚Šï¼Œ%
                \[\angle\text{BCA}=\angle\text{BAS}\]%

                ã¾ãŸï¼Œ$\triangle{\text{ABC}}$ã«ãŠã„ã¦%
                \[\angle\text{ABC}=\angle\text{ACB}+\angle\text{BAC}\]%
                \[\Leftrightarrow\angle\text{ABC}=\angle\text{CAT}\]%

                \ç©ºè¡Œ%
        \end{enumerate}%
        å¾“ã£ã¦å††ã«å†…æŽ¥ã™ã‚‹ä¸‰è§’å½¢ã«ã¤ã„ã¦æˆã‚Šç«‹ã¤ã“ã¨ãŒè¨¼æ˜Žã•ã‚ŒãŸã€‚%
        \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{å†…è§’ã¨å¤–è§’ã®äºŒç­‰åˆ†ç·š}}%
      {%
        \begin{tikzpicture}%
          \draw(5.3,0.8)--(0,0)--(4.8,0)--cycle;%
          \draw(5.3,0.8)--(0,0)--(4,0)--cycle;%
          \draw(5.3,0.8)--(0,0)--(6,0)--cycle;%
          \draw(5.3,0.8)node[above]{A};%
          \draw(0,0)--(6.3,0.950943395);%
          \draw(0,0)node[below]{B};%
          \draw(4.8,0)node[below]{C};%
          \draw(4,0)node[below]{P};%
          \draw(6,0)node[below]{Q};%
          \draw(6.3,0.950943395)node[above]{R};%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\angle\text{BAP}=\angle\text{PAC},\angle\text{CAQ}=\angle\text{QAR}$ã®ã¨ãï¼Œ%
\[\text{BP}:\text{PC}=\text{BQ}:\text{QC}=\text{AB}:\text{AC}\]%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{æ–¹ã¹ãã®å®šç†A}}%
      {%
        \begin{tikzpicture}%
          \draw(0,2)--(1.6,-1.2);%
          \draw(0,2)node[above]{A};%
          \draw(1.6,-1.2)node[below]{B};%
          \draw(-1.6,-1.2)--(1.2,1.6);%
          \draw(-1.6,-1.2)node[below]{C};%
          \draw(1.2,1.6)node[above]{D};%
          \draw(0.7,0.9)node[right]{P};%
          \draw(0,0)circle[radius=2];%
          \draw(0,0)node[below]{O};%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        $\text{PA}\cdot\text{PB}=\text{PC}\cdot\text{PD}$%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{æ–¹ã¹ãã®å®šç†Aã®è¨¼æ˜Ž}}%
      {%
        \è¨¼æ˜Žé–‹å§‹%
        \begin{tikzpicture}%
          \draw(0,2)--(1.6,-1.2);%
          \draw(0,2)node[above]{A};%
          \draw(1.6,-1.2)node[below]{B};%
          \draw(-1.6,-1.2)--(1.2,1.6);%
          \draw(-1.6,-1.2)node[below]{C};%
          \draw(1.2,1.6)node[above]{D};%
          \draw(0.7,0.9)node[right]{P};%
          \draw(0,0)circle[radius=2];%
          \draw(0,0)node[below]{O};%
        \end{tikzpicture}%
        \ç©ºè¡Œ%
        å††å‘¨è§’ã®å®šç†ã‚ˆã‚Šï¼Œ%
\[\angle\text{CAP}=\angle\text{BDP},\quad\angle\text{ACP}=\angle\text{DBP}\]%

        $\triangle{\text{ACP}}\ç›¸ä¼¼\triangle{\text{DBP}}$ã‚ˆã‚Šï¼Œ%
\[\text{PA}\cdot\text{PB}=\text{PC}\cdot\text{PD}\]%
        \è¨¼æ˜Žçµ‚äº†%
      }%
      {\relax}%
    \ifthenelse{\equal{#1}{æ–¹ã¹ãã®å®šç†B}}%
      {%
        \begin{tikzpicture}%
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\[ax^{3}+bx^2+cx+d=a\Ttyuukakko{x-\alpha}\Ttyuukakko{x-\beta}\Ttyuukakko{x-\gamma}=a\Tdaikakko{x^3-\Ttyuukakko{\alpha+\beta+\gamma}x^2+\Ttyuukakko{\alpha\beta+\beta\gamma+\gamma\alpha}x-\alpha\beta\gamma}\]%
\[\Leftrightarrow ax^{3}+bx^2+cx+d=a\Ttyuukakko{x^3+\bunsuu{b}{a}x^2+\bunsuu{c}{a}x+\bunsuu{d}{a}}\]%

        ä¿‚æ•°æ¯”è¼ƒæ³•ã‚ˆã‚Šï¼Œä¸¡è¾ºåŒæ¬¡ã®ä¿‚æ•°ã‚’æ¯”è¼ƒã—ã¦ï¼Œ%
        \[\alpha+\beta+\gamma=-\bunsuu{b}{a}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\alpha\beta+\beta\gamma+\gamma\alpha=\bunsuu{c}{a}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\alpha\beta\gamma=-\bunsuu{d}{a}\]%
        \è¨¼æ˜Žçµ‚äº†%
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\[P\Ttyuukakko{x}=\Ttyuukakko{x-k}Q\Ttyuukakko{x}+R\]%

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        $\text{A}\Ttyuukakko{x_{1}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{1}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ \text{B}\Ttyuukakko{x_{2}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{2}}$ã¨ã—ã¦ï¼Œç·šåˆ†$\text{AB}$é–“ã®è·é›¢ã¯ï¼Œ%
\[\æ&nbsp;¹å·{\Ttyuukakko{x_{2}-x_{1}}^2-\Ttyuukakko{y_{2}-y_{1}}^2}\]%

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        $\text{A}\Ttyuukakko{x_{1}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{1}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ \text{B}\Ttyuukakko{x_{2}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{2}}$ã¨ã—ã¦ï¼Œç·šåˆ†$\text{AB}$ã‚’$m:n$ã«å†…åˆ†ã™ã‚‹ç‚¹ã®åº§æ¨™ã¯ï¼Œ%
\[\Ttyuukakko{\bunsuu{nx_{1}+mx_{2}}{n+m}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{ny_{1}+my_{2}}{n+m}}\]%

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        $m:n$ã«å†…åˆ†ã™ã‚‹ç‚¹ã®åº§æ¨™ã‚’$\text{P}\Ttyuukakko{x\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y}$ã¨ã—ã¦ï¼Œ%
\[m:n=x-x_{1}:x_{2}-x\]%
\[\Leftrightarrow\Ttyuukakko{\bunsuu{nx_{1}+mx_{2}}{n+m}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{ny_{1}+my_{2}}{n+m}}\]%
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        $\text{A}\Ttyuukakko{x_{1}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{1}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ \text{B}\Ttyuukakko{x_{2}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{2}}$ã¨ã—ã¦ï¼Œç·šåˆ†$\text{AB}$ã‚’$m:n$ã«å¤–åˆ†ã™ã‚‹ç‚¹ã®åº§æ¨™ã¯ï¼Œ%
\[\Ttyuukakko{\bunsuu{-nx_{1}+mx_{2}}{m-n}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹  \bunsuu{-ny_{1}+my_{2}}{m-n}}\]%

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\[m:n=x-x_{1}:x-x_{2}\]%
\[\Leftrightarrow\Ttyuukakko{\bunsuu{-nx_{1}+mx_{2}}{m-n}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹  \bunsuu{-ny_{1}+my_{2}}{m-n}}\]%

          \item $m&lt;n$ã®ã¨ã%

                $n:m$ã«å¤–åˆ†ã™ã‚‹ç‚¹ã®åº§æ¨™ã‚’$\text{P}\Ttyuukakko{x\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y}$ã¨ã—ã¦ï¼Œ%
                \[m:n=x-x_{2}:x-x_{1}\]%
                \[\Leftrightarrow\Ttyuukakko{\bunsuu{-nx_{1}+mx_{2}}{m-n}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹  \bunsuu{-ny_{1}+my_{2}}{m-n}}\]%

         \end{enumerate}%
        ã‚ˆã£ã¦$m\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ n$ã®å¤§å°ã«ã‚ˆã‚‰ãš%
         \[\Ttyuukakko{\bunsuu{-nx_{1}+mx_{2}}{m-n}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹  \bunsuu{-ny_{1}+my_{2}}{m-n}}\]%

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        $\text{A}\Ttyuukakko{x_{1}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{1}}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ \text{B}\Ttyuukakko{x_{2}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ y_{2}}$ã¨ã—ã¦ï¼Œç·šåˆ†$\text{AB}$ã®ä¸­ç‚¹ã¯ï¼Œ%
\[\Ttyuukakko{\bunsuu{x_{1}+x_{2}}{2}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{y_{1}+y_{2}}{2}}\]%

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\[\Ttyuukakko{\bunsuu{x_{1}+x_{2}+x_{3}}{3}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹\bunsuu{y_{1}+y_{2}+y_{3}}{3}}\]%

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\[\Leftrightarrow m_{1}=m_{2}\]%

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\[\Leftrightarrow m_{1}m_{2}=-1\]%

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    \ifthenelse{\equal{#1}{å…¬å¼Bã®è¨¼æ˜Ž}}%
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        \è¨¼æ˜Žé–‹å§‹%
        $y=mx_{1}$ä¸Šã«ç‚¹A$\Ttyuukakko{1\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ m_{1}}$\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ $y=mx_{2}$ä¸Šã«B$\Ttyuukakko{-m_{1}\æ•°å¼ã‚«ãƒ³ãƒžã‚¹ãƒšãƒ¼ã‚¹ 1}$ã‚’ã¨ã‚‹ã€‚%

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      {%
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                  \Leftrightarrow d&amp;=\æ&nbsp;¹å·{\Tdaikakko{\Ttyuukakko{\bunsuu{-a\Ttyuukakko{ax_{1}+by_{1}+c}}{a^2+b^2}}^2+\Tdaikakko{\bunsuu{-b\Ttyuukakko{ax_{1}+by_{1}+c}}{a^2+b^2}}}^2}&amp;\\%
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          \item $a=0$ã®ã¨ã%

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\[\bunsuu{\Tzettaiti{ax_{1}+by_{2}+c}}{\æ&nbsp;¹å·{a^2+b^2}}\]%

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      {$\cos\Ttyuukakko{\alpha\pm\beta}=\cos\alpha \cos\beta\mp \sin\alpha \sin\beta\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}$}{\relax}%
    \ifthenelse{\equal{#1}{å…¬å¼B}\AND\equal{#2}{b}}%
      {\[\cos\Ttyuukakko{\alpha\pm\beta}=\cos\alpha \cos\beta\mp \sin\alpha \sin\beta\]\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}}{\relax}%
    \ifthenelse{\equal{#1}{å…¬å¼C}\AND\equal{#2}{i}}%
      {$\tan\Ttyuukakko{\alpha\pm\beta}=\bunsuu{\tan\alpha \pm \tan\beta}{1\mp \tan\alpha \tan\beta}\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}$}{\relax}%
    \ifthenelse{\equal{#1}{å…¬å¼C}\AND\equal{#2}{b}}%
      {\[\tan\Ttyuukakko{\alpha\pm\beta}=\bunsuu{\tan\alpha \pm \tan\beta}{1\mp \tan\alpha \tan\beta}\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}\]}{\relax}%
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        \è¨¼æ˜Žé–‹å§‹%
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          \draw(0,0)--(0.75,1);%
          \draw(0,0)node[below right]{O};%
          \draw(0.8,1)node[right]{P$\Ttyuukakko{\cos\alpha,\sin\alpha}$};%
          \draw(0,0)circle[radius=1.25];%
          \draw(0,-1.25)node[below left]{$-1$};%
          \draw(-1.25,0)node[below left]{$-1$};%
          \draw(0,1.25)node[above left]{$1$};%
          \draw(1.25,0)node[below right]{$1$};%
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        å›³ã«ãŠã„ã¦ï¼Œä¸‰è§’é–¢æ•°ã®æ€§è³ªã‚ˆã‚Š$\cos\Ttyuukakko{\beta-\alpha}=\cos\Ttyuukakko{\alpha-\beta}$ãªã®ã§ï¼Œ$\triangle{\text{QOP}}$ã«ã¤ã„ã¦ä½™å¼¦å®šç†ã‚ˆã‚Š%
\[\text{QP}^2=1^2+1^2-2\cdot1\cdot1\cdot\cos\Ttyuukakko{\alpha-\beta}\]%

        ã¾ãŸï¼ŒQPé–“ã®è·é›¢ã«ã¤ã„ã¦ä¸‰å¹³æ–¹ã®å®šç†ã‚’ç”¨ã„ã¦%
\[\text{QP}^2=\Ttyuukakko{\cos\beta-\cos\alpha}^2+\Ttyuukakko{\sin\alpha-\sin\beta}^2\]%
\[\Leftrightarrow 2-2\cos\Ttyuukakko{\alpha-\beta}=\Ttyuukakko{\cos\beta-\cos\alpha}^2+\Ttyuukakko{\sin\alpha-\sin\beta}^2\]%

        ä¸¡è¾ºæ•´ç†ã—ã¦ï¼Œ%
\[\cos\Ttyuukakko{\alpha-\beta}=\cos\alpha\cos\beta+\sin\alpha\sin\beta\]%

        ã‚’å¾—ã‚‹ã€‚%

        ã¾ãŸï¼Œ$\sin-\theta=-\sin\theta$ã‚ˆã‚Šï¼Œ%
\[\cos\Ttyuukakko{\alpha+\beta}=\cos\alpha\cos\beta-\sin\alpha\sin\beta\]%

        \ç©ºè¡Œ%
\[\cos\Ttyuukakko{\alpha-\beta}=\cos\alpha\cos\beta+\sin\alpha\sin\beta\]%

        ã«ãŠã„ã¦ï¼Œ$\alpha$ã‚’$\bunsuu{\pi}{2}-\alpha$ã«ã™ã‚‹ã“ã¨ã§ï¼Œ%
\[\sin\Ttyuukakko{\alpha+\beta}=\sin\alpha \cos\beta+ \cos\alpha \sin\beta\]%

        ã“ã“ã§ï¼Œ$\beta$ã‚’$-\beta$ã«ã™ã‚‹ã“ã¨ã§ï¼Œ%
\[\sin\Ttyuukakko{\alpha-\beta}=\sin\alpha \cos\beta-\cos\alpha \sin\beta\]%

        \ç©ºè¡Œ%
        $\tan\theta=\bunsuu{\sin\theta}{\cos\theta}$ã‚ˆã‚Šï¼Œ%
\[\tan\Ttyuukakko{\alpha\pm\beta}=\bunsuu{\sin\alpha \cos\beta\pm \cos\alpha \sin\beta}{\cos\alpha \cos\beta\mp \sin\alpha \sin\beta}\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}\]%

        ä¸¡è¾ºã‚’$\cos\alpha\cos\beta$ã§ã‚ã‚‹ã“ã¨ã§ï¼Œ%
\[\tan\Ttyuukakko{\alpha\pm\beta}=\bunsuu{\tan\alpha \pm \tan\beta}{1\mp \tan\alpha \tan\beta}\text{\ ï¼ˆè¤‡å·åŒé&nbsp;†ï¼‰}\]%

        ã‚’å¾—ã‚‹ã€‚%
        \è¨¼æ˜Žçµ‚äº†%
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            $\sin\Ttyuukakko{\alpha+\beta}=\sin\alpha \cos\beta+\cos\alpha \sin\beta$\\%
            $\cos\Ttyuukakko{\alpha+\beta}=\cos\alpha \cos\beta-\sin\alpha \sin\beta$\\%
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        \hspace{3\zw}\phrasesmath[c]%
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        ã‚’å¾—ã‚‹ã€‚%

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\[\cos2\theta=2\cos^{2}\theta-1\]%
\[\Leftrightarrow\cos2\theta=1-2\sin^{2}\theta\]%

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        ä¸‰è§’é–¢æ•°ã®åŠ&nbsp;æ³•å®šç†%
\[\sin\Ttyuukakko{\alpha+\beta}=\sin\alpha \cos\beta+ \cos\alpha \sin\beta\]%
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\[\sin3\theta=\sin\theta \cos2\theta+ \cos\theta \sin2\theta\]%
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        äºŒå€è§’ã®å…¬å¼ã¨ä¸‰è§’é–¢æ•°ã®ç›¸äº’é–¢ä¿‚ã‚ˆã‚Šï¼Œ%
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    \ifthenelse{\equal{#1}{å…¬å¼C}\AND\equal{#2}{b}}%
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        \è¨¼æ˜Žé–‹å§‹%
        ä¸‰è§’é–¢æ•°ã®åŠ&nbsp;æ³•å®šç†%
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          \sin\Ttyuukakko{\alpha-\beta}=\sin\alpha\cos\beta-\cos\alpha\sin\beta\shikimaru{2}\\%
          \cos\Ttyuukakko{\alpha+\beta}=\cos\alpha\cos\beta-\sin\alpha\sin\beta\shikimaru{3}\\%
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        \end{align*}
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        \åŠç©ºè¡Œ%
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        \åŠç©ºè¡Œ%
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\[\sin\alpha\sin\beta=\bunsuu{1}{2}\Tdaikakko{\cos\Ttyuukakko{\alpha-\beta}-\cos\Ttyuukakko{\alpha+\beta}}\]%
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        \è¨¼æ˜Žé–‹å§‹%
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\[\sin\alpha\cos\beta=\bunsuu{1}{2}\Tdaikakko{\sin\Ttyuukakko{\alpha+\beta}+\sin\Ttyuukakko{\alpha-\beta}}\]%
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