X-Git-Url: https://melusine.eu.org/syracuse/G/git/?p=mp-solid.git;a=blobdiff_plain;f=doc%2Fchrist5.tex;fp=doc%2Fchrist5.tex;h=4cad82590b3958570d51e2bc4e86b203c99ce48d;hp=0000000000000000000000000000000000000000;hb=2d234af911ec062a36b45701b19b1065c765eb4e;hpb=9100d101b0c07b6c3510d76f95ec09f68831ca9d

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+%=========================================
+%Macros personnelles
+%christophe.poulain@melusine.eu.org
+%création : 25 Septembre 1999
+%dernière modification : 28 Avril 2005
+%=========================================
+
+\RequirePackage{xlop,fancybox,color,amssymb,ifthen}
+\input{xlopsqrt}
+\newtheorem{ppte}{Propri\'et\'e}
+\newtheorem{theo}{Th\'eor\`eme}
+\newtheorem{defi}{D\'efinition}
+\newtheorem{lemme}{Lemme}
+\newtheorem{coro}{Corollaire}
+\newtheorem{prop}{Proposition}
+\newtheorem{reg}{R\`egle}
+\newtheorem{conj}{Conjecture}
+\newtheorem{remar}{Remarque}
+\newtheorem{exem}{Exemple}
+
+\newcommand{\rema}{\underline{Remarque} }
+\newcommand{\exe}{\underline{Exemple} }
+\newcommand{\pre}{\underline{Preuve}}
+\newcommand{\cas}{\underline{Cas particulier}}
+\newcommand{\cass}{\underline{Cas particuliers}}
+\newcommand{\Not}{\underline{Notation} }
+\newcommand{\Si}{\underline{Si} }
+\newcommand{\si}{\underline{si} }
+\newcommand{\alors}{\underline{alors} }
+\newcommand{\cons}{\underline{Conséquence}}
+\newcommand{\Comme}{\underline{Comme} }
+\newcommand{\comme}{\underline{comme} }
+
+\def\qed{\hfill\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width4pt\vfill\hrule}\vrule}}
+\def\cqfd{\hfill\unskip\kern 6pt\penalty 500\qed\par}
+
+\catcode`\@=11
+\def\Eqalign#1{\null\,\vcenter{\openup\jot\m@th\ialign{
+\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
+&&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$
+\hfil\crcr #1\crcr}}\,}
+\catcode`\@=12
+
+\newcommand{\vecteur}[1]
+{\overrightarrow{\strut #1}}
+
+\font\tenbb=msbm10
+\font\sevenbb=msbm7
+\font\fivebb=msbm5
+\newfam\bbfam
+\textfont\bbfam=\tenbb
+\scriptfont\bbfam=\sevenbb
+\scriptscriptfont\bbfam=\fivebb
+\def\bb{\fam\bbfam\tenbb}
+\let\oldbb=\bb
+\def\bb #1{{\oldbb #1}}
+
+\def\tvi{\vrule height 12pt depth 5pt width 0pt}
+\def\tvj{\vrule height 12pt depth 5pt width 1pt}
+\def\hfq{\hfill\,\,}
+\def\cc#1{\hfq #1\hfq}
+\def\tv{\tvi\vrule}
+\def\tw{\tvj\vrule}
+\def\traithorizontal{\noalign{\hrule}}
+\def\traithorizontale{\noalign{\hrule height 1pt}}
+
+\newcommand{\encadre}[1]
+{\begin{center}
+\fbox{\begin{minipage}{\linewidth}
+{#1}
+\end{minipage}}
+\end{center}
+}
+
+\def\pgcd{\mathop{\rm PGCD}\nolimits}
+\def\ppcm{\mathop{\rm PPCM}\nolimits}
+
+\def\cut{{}\hfill\cr \hfill{}}
+
+\newcommand{\biindice}[3]%
+{
+\renewcommand{\arraystretch}{0.5}
+\begin{array}[t]{c}
+#1\\
+{\scriptstyle #2}\\
+{\scriptstyle #3}
+\end{array}
+\renewcommand{\arraystretch}{1}
+}
+
+\newlength{\ltxt}
+\newcommand{\compo}[4]{
+\setlength{\ltxt}{\linewidth}
+\setbox#1=\hbox{\includegraphics[scale=#3]{#2.#1}}
+\addtolength{\ltxt}{-\wd#1}
+\addtolength{\ltxt}{-10pt}
+\begin{minipage}{\wd#1}
+\includegraphics[scale=#3]{#2.#1}
+\end{minipage}
+\hfill
+\begin{minipage}{\ltxt}
+#4
+\end{minipage}
+}
+
+\newcommand{\compog}[4]{
+\setlength{\ltxt}{\linewidth}
+\setbox#1=\hbox{\includegraphics[scale=#3]{#2.#1}}
+\addtolength{\ltxt}{-\wd#1}
+\addtolength{\ltxt}{-10pt}
+\begin{minipage}{\ltxt}
+#4
+\end{minipage}
+\hfill
+\begin{minipage}{\wd#1}
+\includegraphics[scale=#3]{#2.#1}
+\end{minipage}
+}
+
+\newlength{\lntxt}
+\newcommand{\Compo}[4]{
+\setlength{\lntxt}{\linewidth}
+\setbox#1=\hbox{\includegraphics[scale=#3]{#2}}
+\addtolength{\lntxt}{-\wd#1}
+\addtolength{\lntxt}{-10pt}
+\begin{minipage}{\wd#1}
+\includegraphics[scale=#3]{#2}
+\end{minipage}
+\hfill
+\begin{minipage}{\lntxt}
+#4
+\end{minipage}
+}
+
+\newlength{\lnttxt}
+\newcommand{\dispo}[3]{
+\setlength{\lnttxt}{\linewidth}
+\setbox#1=\hbox{#2}
+\addtolength{\lnttxt}{-\wd#1}
+\addtolength{\lnttxt}{-20pt}
+\begin{minipage}{\wd#1}
+#2
+\end{minipage}
+\hfill
+\begin{minipage}{\lnttxt}
+#3
+\end{minipage}
+}
+
+\newcounter{num}[section]
+\newcommand{\exo}{\addtocounter{num}{1}
+\par\underline{\bf Exercice~\thenum} }
+
+\newcommand{\titrage}[2]{
+{\Large #1}\hfill#2
+\par\rule[+6pt]{\linewidth}{0.5mm}
+\par
+}
+
+\newcommand{\titragedossier}[1]{
+{\small #1}\hfill{\small www.melusine.eu.org/syracuse/poulecl/}
+\par\rule[+6pt]{\linewidth}{0.5mm}
+\par
+}
+
+\newcommand{\partie}[2]{
+\begin{center}
+\begin{minipage}{#1pt}
+\begin{center}
+\boxput*(0,0){\colorbox{white}{#2}}
+{\rule{\linewidth}{0.5mm}}
+\end{center}
+\end{minipage}
+\end{center}
+\par
+}
+
+\newenvironment{myenumerate}{
+\renewcommand{\theenumi}{\arabic{enumi}}
+\def\labelenumi{{\bf \theenumi /}}
+\begin{enumerate}}{\end{enumerate}}
+
+\newenvironment{Myenumerate}{
+\renewcommand{\theenumi}{\arabic{enumi}}
+\def\labelenumi{$\rhd${\bf \theenumi /}}
+\begin{enumerate}}{\end{enumerate}}
+
+\newdimen\shadeshift\shadeshift=1pt
+\def\shadedtext#1{{\setbox0=\hbox{#1}\leavevmode
+        \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}%
+        \kern0.1em\lower0.1em\copy0
+        \special{color pop}}\vss}\box0}}%
+\long\def\shadedparagraph#1\par{{\setbox0=\vbox{\hsize=\hsize#1}%
+        \noindent\leavevmode
+        \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}%
+        \kern0.1em\lower0.1em\copy0
+        \special{color pop}}\vss}\box0\par}}%
+
+\newboolean{exact}
+\setboolean{exact}{true}
+\newboolean{racine}
+\setboolean{racine}{false}
+
+\newcommand{\pythahypo}[5]{
+\opset{decimalsepsymbol={,}}
+\opcopy{#4}{A1}
+\opcopy{#5}{A2}
+Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire :
+\[\Eqalign{
+#1#3^2&=#1#2^2+#2#3^2\cr
+#1#3^2&=\opprint{A1}^2+\opprint{A2}^2\cr
+#1#3^2&=\opmul*{A1}{A1}{a1}\opprint{a1}+\opmul*{A2}{A2}{a2}\opprint{a2}\cr
+#1#3^2&=\opadd*{a1}{a2}{a3}\opprint{a3}\cr
+#1#3&=\sqrt{\opprint{a3}}\cr
+\ifthenelse{\boolean{racine}}{}{\ifthenelse{\boolean{exact}}{#1#3&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#3&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4}
+\opprint{a4}}}\cr
+}\]
+}
+
+\newcommand{\pythadroit}[5]{
+\opset{decimalsepsymbol={,}}
+\opcopy{#4}{A1}
+\opcopy{#5}{A2}
+Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire :
+\[\Eqalign{
+#1#3^2&=#1#2^2+#2#3^2\cr
+\opprint{A1}^2&=#1#2^2+\opprint{A2}^2\cr
+\opmul*{A1}{A1}{a1}\opprint{a1}&=#1#2^2+\opmul*{A2}{A2}{a2}\opprint{a2}\cr
+#1#2^2&=\opmul*{A1}{A1}{a1}\opprint{a1}-\opmul*{A2}{A2}{a2}\opprint{a2}\cr
+#1#2^2&=\opsub*{a1}{a2}{a3}\opprint{a3}\cr
+#1#2&=\sqrt{\opprint{a3}}\cr
+\ifthenelse{\boolean{racine}}{}{\ifthenelse{\boolean{exact}}{#1#2&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#2&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4}
+\opprint{a4}}}\cr
+}\]
+}
+
+\newcommand{\Recipytha}[6]{
+\opset{decimalsepsymbol={,}}
+\opcopy{#4}{A1}
+\opcopy{#5}{A2}
+\opcopy{#6}{A3}
+Dans le triangle $#1#2#3$, $[#1#3]$ est le plus grand côté.
+\[\left.
+  \begin{array}{l}
+    #1#3^2=\opprint{A1}^2=\opmul*{A1}{A1}{a1}\opprint{a1}\cr
+    #1#2^2+#2#3^2=\opprint{A2}^2+\opprint{A3}^2=\opmul*{A2}{A2}{a2}\opprint{a2}+\opmul*{A3}{A3}{a3}\opprint{a3}=\opadd*{a2}{a3}{a4}\opprint{a4}\cr
+  \end{array}
+\right\}#1#3^2=#1#2^2+#2#3^2
+\]
+Comme $#1#3^2=#1#2^2+#2#3^2$ alors le triangle $#1#2#3$ est rectangle en $#2$ d'après la réciproque du théorème de Pythagore.}
+
+\newcommand{\Recipythacol}[6]{
+\opset{decimalsepsymbol={,}}
+\opcopy{#4}{A1}
+\opcopy{#5}{A2}
+\opcopy{#6}{A3}
+Dans le triangle $#1#2#3$, $[#1#3]$ est le plus grand côté.
+\[\Eqalign{
+    #1#3^2&\kern0.15\linewidth&#1#2^2+#2#3^2\cr
+    \opprint{A1}^2&&\opprint{A2}^2+\opprint{A3}^2\cr
+    \opmul*{A1}{A1}{a1}\opprint{a1}&&\opmul*{A2}{A2}{a2}\opprint{a2}+\opmul*{A3}{A3}{a3}\opprint{a3}\cr
+    &&\opadd*{a2}{a3}{a4}\opprint{a4}\cr
+}\]
+Comme $#1#3^2=#1#2^2+#2#3^2$ alors le triangle $#1#2#3$ est rectangle en $#2$ d'après la réciproque du théorème de Pythagore.}
+
+\newcommand{\Thales}[5]{%
+Dans le triangle $#1#2#3$, $#4$ est un point de la
+ droite $(#1#2)$, $#5$ est un point de la droite
+ $(#1#3)$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles.
+ Le théorème de Thalès permet d'écrire :
+\[\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}\]%
+}
+
+\newcommand{\Thalesf}[5]{
+Dans le triangle $#1#2#3$, $#4$ est un point du
+ segment $[#1#2]$, $#5$ est un point du segment
+ $[#1#3]$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles.
+ L'égalité des 3 rapports permet d'écrire :
+\[\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}\]
+}
+
+\newcommand{\ResolThales}[6]{%
+\opset{decimalsepsymbol={,}}%
+\opcopy{#3}{a3}%
+\opcopy{#4}{a4}%
+\opcopy{#5}{a5}%
+On utilise %
+\[\Eqalign{%
+\frac{#1#2}{\opprint{a3}}&=\frac{\opprint{a4}}{\opprint{a5}}\cr%
+#1#2&=\frac{\opprint{a3}\times\opprint{a4}}{\opprint{a5}}\cr%
+#1#2&=\frac{\opmul*{a3}{a4}{a6}\opunzero{a6}\opprint{a6}}{\opprint{a5}}\cr%
+\ifthenelse{\boolean{exact}}{#1#2&=\opdiv*[maxdivstep=3]{a6}{a5}{a7}{a8}\opunzero{a7}\opprint{a7}\cr}{#1#2&\approx\opdiv*[maxdivstep=3]{a6}{a5}{a7}{a8}\opunzero{a7}\opprint{a7}\cr}%
+}\]%
+\ifthenelse{\boolean{exact}}{La longueur $#1#2$ mesure \opprint{a7}\,#6}{La longueur $#1#2$ mesure environ \opprint{a7}\,#6}%
+}
+
+%%QCM
+\newcounter{qqcm}
+
+%définir un booléen qui permet de choisir la correction ou non
+\newboolean{solution}
+
+%définir une commande \V qui permet de changer le carré en carré coché suivant la valeur du booléen.
+\newcommand{\V}[1]{\ifthenelse{\boolean{solution}}{$\boxtimes$\kern2mm #1}{$\Box$\kern2mm #1}}
+\newcommand{\F}[1]{$\Box$\kern2mm #1}
+\newcommand{\vr}{\ifthenelse{\boolean{solution}}{$\boxtimes$}{$\Box$}}
+\newcommand{\fa}{$\Box$}
+
+%%QCM Version 2
+\newenvironment{Qcm}[1][2]{\par\setboolean{solution}{false}
+\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular}
+\renewcommand{\arraystretch}{1}}
+
+\newenvironment{Qcmcor}[1][2]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular}
+\renewcommand{\arraystretch}{1}}
+
+\newcounter{taill}
+\newcommand{\QCM}[3]{\setboolean{solution}{false}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{1.5}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+\multicolumn{\thetaill}{|c|}{{\sc #2}}\\
+#3
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newcommand{\QCMcor}[3]{\setboolean{solution}{true}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{1.5}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+\multicolumn{\thetaill}{|c|}{{\sc #2}}\\
+#3
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newcommand{\QCMvar}[4]{\setboolean{solution}{false}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{#2}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+\multicolumn{\thetaill}{|c|}{{\sc #3}}\\
+#4
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newcommand{\QCMvarcor}[4]{\setboolean{solution}{true}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{#2}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+\multicolumn{\thetaill}{|c|}{{\sc #3}}\\
+#4
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newcommand{\QCMsimple}[2]{\setboolean{solution}{false}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{1.5}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+#2
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newcommand{\QCMsimplevar}[3]{\setboolean{solution}{false}
+\setcounter{qqcm}{0}
+\renewcommand{\arraystretch}{#2}
+\setcounter{taill}{#1}
+\addtocounter{taill}{1}
+\begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
+\hline
+#3
+\hline
+\end{tabularx}
+\renewcommand{\arraystretch}{1}
+}
+
+\newenvironment{VF}[1]{\par\setboolean{solution}{false}
+\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
+\begin{center}
+\begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline
+\multicolumn{3}{|c|}{#1}\\
+\hline
+\multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\
+\hline
+}{\hline\end{tabular}
+\end{center}\renewcommand{\arraystretch}{1}}
+
+\newenvironment{VFcor}[1]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline
+\multicolumn{3}{|c|}{#1}\\
+\hline
+\multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\
+\hline
+}{\hline\end{tabular}
+\renewcommand{\arraystretch}{1}}