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% Title Page
\title{Thalès - Travail de groupe}
\author{}
\date{}

\fancyhead[L]{Quatrième}
\fancyhead[C]{\Thetitle}
\fancyhead[R]{\thepage}


\begin{document}
\thispagestyle{fancy}

Dans tous les dessins suivants, $ABC$ et $AMN$ sont deux triangles tels que $M$ est un point de $[AB]$, $N$ un point de $[AC]$ et $(MN)//(BC)$. Dans chacunes des figures, les longueurs de quatres côtés sont données. (Les mesures sur les dessins ne sont pas respectées)

\textbf{Conjecturer les valeurs des longueurs des deux cotés manquants pour remplir les tableaux des longueurs.} Justifier quand c'est possible les valeurs trouvées. 

\begin{itemize}
    \item \textbf{Cas 1}

        \begin{tikzpicture}
                    \coordinate (A) at (0,0);
                    \coordinate (M) at (3,0);
                    \coordinate (B) at (6,0);
                    \coordinate (C) at (6,4);
                    \coordinate (N) at (3,2);

                    \draw (A) node [left] {$A$}
                    -- (M) node [below] {$M$} node[midway, below] {$4$}
                    -- (B) node [right] {$B$} 
                    -- (C) node [right] {$C$} 
                    -- (N) node [above] {$N$} 
                    -- (A) node [midway, sloped, above] {5};

                    \draw (N) -- (M) node[midway, right] {3};
                    \draw[<->] ($(A)+(0,-0.5)$) -- ($(B) + (0,-0.5)$) node [midway, below] {$8$};
            
        \end{tikzpicture}

        \begin{tabular}{|c|*{3}{p{2cm}|}}
            \hline
            Triangle $AMN$ & AM =  & AN =  & MN =  \\
            \hline
            Triangle $ABC$ & AB =  & AC =  & BC =  \\
            \hline
        \end{tabular}
    \item \textbf{Cas 2}

        \begin{tikzpicture}
                    \coordinate (A) at (0,0);
                    \coordinate (M) at (2.3,0);
                    \coordinate (B) at (9.2,0);
                    \coordinate (C) at (8,4);
                    \coordinate (N) at (2,1);

                    \draw (A) node [left] {$A$}
                    -- (M) node [below] {$M$} node[midway, below] {$1$}
                    -- (B) node [right] {$B$} 
                    -- (C) node [right] {$C$} 
                    -- (N) node [above] {$N$} 
                    -- (A) node [midway, sloped, above] {5};

                    \draw (N) -- (M) node[midway, right] {3};
                    \draw[<->] ($(A)+(0,-0.5)$) -- ($(B) + (0,-0.5)$) node [midway, below] {4};
                    \draw (A) --++ (2.3,0) --++ (2.3,0) node {$|$} --++ (2.3,0) node {$|$};
            
        \end{tikzpicture}

        \begin{tabular}{|c|*{3}{p{2cm}|}}
            \hline
            Triangle $AMN$ & AM =  & AN =  & MN =  \\
            \hline
            Triangle $ABC$ & AB =  & AC =  & BC =  \\
            \hline
        \end{tabular}
    \item \textbf{Cas 3}

        \begin{tikzpicture}
                    \coordinate (A) at (0,0);
                    \coordinate (M) at (3,0);
                    \coordinate (B) at (6,0);
                    \coordinate (C) at (6.8,4);
                    \coordinate (N) at (3.4,2);

                    \draw (A) node [left] {$A$}
                    -- (M) node [below] {$M$} node[midway, below] {$4$}
                    -- (B) node [right] {$B$} 
                    -- (C) node [right] {$C$} 
                    -- (N) node [above] {$N$} 
                    -- (A) node [midway, sloped, above] {5};

                    \draw (N) -- (M) node[midway, right] {4.4};
                    \draw[<->] ($(A)+(-0.5,+0.5)$) -- ($(C) + (0,+0.75)$) node [midway, above, sloped] {$10$};
            
        \end{tikzpicture}

        \begin{tabular}{|c|*{3}{p{2cm}|}}
            \hline
            Triangle $AMN$ & AM =  & AN =  & MN =  \\
            \hline
            Triangle $ABC$ & AB =  & AC =  & BC =  \\
            \hline
        \end{tabular}
    \item \textbf{Cas 4}

        \begin{tikzpicture}
                    \coordinate (A) at (0,0);
                    \coordinate (M) at (3,0);
                    \coordinate (B) at (6,0);
                    \coordinate (C) at (6,4);
                    \coordinate (N) at (3,2);

                    \draw (A) node [left] {$A$}
                    -- (M) node [below] {$M$} node[midway, below] {$4$}
                    -- (B) node [right] {$B$} 
                    -- (C) node [right] {$C$} node [midway, right] {15}
                    -- (N) node [above] {$N$} 
                    -- (A) node [midway, sloped, above] {5};

                    \draw (N) -- (M) node[midway, right] {3};
            
        \end{tikzpicture}

        \begin{tabular}{|c|*{3}{p{2cm}|}}
            \hline
            Triangle $AMN$ & AM =  & AN =  & MN =  \\
            \hline
            Triangle $ABC$ & AB =  & AC =  & BC =  \\
            \hline
        \end{tabular}
\end{itemize}

\end{document}

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