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\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{colortbl}
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\usepackage{fancyhdr}

% French language support
\usepackage[french]{babel}

% Bottom message in Arabic requires bidi package
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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 03 - S01}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}


% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings



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% Document spacing
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% Math display style
\everymath{\displaystyle}

%\mathversion{bold}

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% Custom theorem environments (uncomment if needed)
% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma}
% \newtheorem{corollary}{Corollary}

% Custom exercise environments (uncomment if needed)
% \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
% \newenvironment{remark}{\begin{proof}[Remark]}{\end{proof}}

% Custom commands for this exam (add your own below)
% \newcommand{\answerline}[1]{\underline{\hspace{\#1}}}
% \newcommand{\points}[1]{\hfill(\mbox{\#1 points})}

% Additional packages (uncomment if needed)
% \usepackage{siunitx} % For SI units
% \usepackage{chemformula} % For chemical formulas
% \usepackage{circuitikz} % For circuit diagrams

% Page layout adjustments
% \setlength{\parindent}{0pt}
% \setlength{\parskip}{1em}

% Custom colors for this exam
% \definecolor{myblue}{RGB}{0,100,200}
% \definecolor{mygreen}{RGB}{0,150,0}

% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
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\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}

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\newcommand{\printexo}[3]{%
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    \def\fulltitle{Exercise #1~#2}%
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% --- Header Style 9 (fr) ---
\newcommand{\printheadnine}{%
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      \wsite & \centering \large\textbf{\secondtitle}
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\begin{document}



\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{}{
Soit \(ABC\) un triangle rectangle en \(A\) tels que \(AB = 6\ \text{cm}\) et \(AC = 3\ \text{cm}\).\\
Soit \(G = \text{Bary}\{(A;1);(B;3);(C;-1)\}\) et \(E = \text{Bary}\{(A;1);(B;4);(C;-2)\}\).
\begin{enumerate}[label={\arabic*)}]
    \item
    \begin{enumerate}
        \item Écrire \(\overrightarrow{AG}\) et \(\overrightarrow{AE}\) en fonction de \(\overrightarrow{AB}\) et \(\overrightarrow{AC}\).
        \item Construire une figure convenable.
    \end{enumerate}
    \item Montrer que \((GE) // (BC)\).
    \item Soit \(I\) un point tel que \(5\overrightarrow{AI} = 4\overrightarrow{AB}\).
    \begin{enumerate}[label={a-}]
        \item Montrer que \(I = \text{Bary}\{(A;1);(B;4)\}\).
        \item En déduire que les points \(A, B\) et \(I\) sont alignés.
    \end{enumerate}
    \item La droite \((AE)\) coupe la droite \((BC)\) au point \(J\). Montrer que \(J = \text{Bary}\{(B;\alpha);(C;\beta)\}\) où l’on déterminera les valeurs de \(\alpha\) et \(\beta\).
    \item Déterminer l’ensemble des points \(M\) tel que :
    ~\( \|\overrightarrow{MA} + 4\overrightarrow{MB} - 2\overrightarrow{MC}\| = \|\overrightarrow{MA} + \overrightarrow{MB} - \overrightarrow{MC}\| \)~
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Le plan est rapporté à un repère orthonormé $(O; \vec i, \vec j)$\\
On considère les points : \(A(3;2)\), \(B(5;6)\) et \(C(-1;4)\).
\begin{enumerate}[label={\arabic*)}, tight]
  \item Calculer \(\cos(\overrightarrow{BA},\overrightarrow{BC})\) et \(\sin(\overrightarrow{BA},\overrightarrow{BC})\) puis en déduire la mesure principale de l’angle \((\widehat{\overrightarrow{BA},\overrightarrow{BC}})\).
    \item
    \begin{enumerate}[label={a-}]
        \item Déterminer les équations cartésiennes des hauteurs \((D)\) et \((\Delta)\) du triangle \(ABC\) issues respectivement de \(A\) et \(B\).
        \item En déduire les coordonnées du centre de gravité du triangle \(ABC\).
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
Dans le plan \((P)\), on considère les points \(A(2;0), B(-1;0)\) et \(C(1;3)\). Soit \((C_k)\) un ensemble défini par :
~\( (C_k) = \{M(x;y) \in (P): 4AM^2 - BM^2 = 3k\} \)~
tel que \(k \in \mathbb{R}\).

\begin{enumerate}[label={\arabic*)}]
    \item Vérifier que pour tout \(M(x;y) \in (P)\) on a \(4AM^2 - BM^2 = 3(x^2 + y^2 - 6x + 5)\).
    \item Montrer que l’ensemble \((C_0)\) est un cercle dont on déterminera son centre et son rayon.
    \item Étudier suivant les valeurs du paramètre \(k\) la nature de l’ensemble \((C_k)\).
    \item Quelle est la valeur de \(k\) pour que \(C \in (C_k)\) et donner dans ce cas l’équation de la tangente \((\Delta)\) de l’ensemble \((C_k)\) au point \(C\).
    \item
    \begin{enumerate}[label={a-}]
        \item Vérifier que le point \(O\) est à l’extérieur du cercle \((C_0)\).
        \item Déterminer les équations des deux tangentes \((T_1)\) et \((T_2)\) au cercle \((C_0)\) issues de \(O\).
    \end{enumerate}
    \item Résoudre graphiquement le système \((S)\) :
    ~\(\begin{cases}
        x^2+y^2-6x+5>0\\
        \sqrt{5}|y| < 2x
    \end{cases} \)~
\end{enumerate}
}





% Bottom message
\vspace*{-0.5cm}
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ الٓمٓر ۚ تِلْكَ ءَايَـٰتُ ٱلْكِتَـٰبِ ۗ وَٱلَّذِىٓ أُنزِلَ إِلَيْكَ مِن رَّبِّكَ ٱلْحَقُّ وَلَـٰكِنَّ أَكْثَرَ ٱلنَّاسِ لَا يُؤْمِنُونَ﴾ (الرعد 1)
    }%
  }%
\end{center}

\end{document}