doc/christ5.tex

   1 %=========================================
   2 %Macros personnelles
   3 %christophe.poulain@melusine.eu.org
   4 %création : 25 Septembre 1999
   5 %dernière modification : 28 Avril 2005
   6 %=========================================
   7
   8 \RequirePackage{xlop,fancybox,color,amssymb,ifthen}
   9 \input{xlopsqrt}
  10 \newtheorem{ppte}{Propri\'et\'e}
  11 \newtheorem{theo}{Th\'eor\`eme}
  12 \newtheorem{defi}{D\'efinition}
  13 \newtheorem{lemme}{Lemme}
  14 \newtheorem{coro}{Corollaire}
  15 \newtheorem{prop}{Proposition}
  16 \newtheorem{reg}{R\`egle}
  17 \newtheorem{conj}{Conjecture}
  18 \newtheorem{remar}{Remarque}
  19 \newtheorem{exem}{Exemple}
  20
  21 \newcommand{\rema}{\underline{Remarque} }
  22 \newcommand{\exe}{\underline{Exemple} }
  23 \newcommand{\pre}{\underline{Preuve}}
  24 \newcommand{\cas}{\underline{Cas particulier}}
  25 \newcommand{\cass}{\underline{Cas particuliers}}
  26 \newcommand{\Not}{\underline{Notation} }
  27 \newcommand{\Si}{\underline{Si} }
  28 \newcommand{\si}{\underline{si} }
  29 \newcommand{\alors}{\underline{alors} }
  30 \newcommand{\cons}{\underline{Conséquence}}
  31 \newcommand{\Comme}{\underline{Comme} }
  32 \newcommand{\comme}{\underline{comme} }
  33
  34 \def\qed{\hfill\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width4pt\vfill\hrule}\vrule}}
  35 \def\cqfd{\hfill\unskip\kern 6pt\penalty 500\qed\par}
  36
  37 \catcode`\@=11
  38 \def\Eqalign#1{\null\,\vcenter{\openup\jot\m@th\ialign{
  39 \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
  40 &&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$
  41 \hfil\crcr #1\crcr}}\,}
  42 \catcode`\@=12
  43
  44 \newcommand{\vecteur}[1]
  45 {\overrightarrow{\strut #1}}
  46
  47 \font\tenbb=msbm10
  48 \font\sevenbb=msbm7
  49 \font\fivebb=msbm5
  50 \newfam\bbfam
  51 \textfont\bbfam=\tenbb
  52 \scriptfont\bbfam=\sevenbb
  53 \scriptscriptfont\bbfam=\fivebb
  54 \def\bb{\fam\bbfam\tenbb}
  55 \let\oldbb=\bb
  56 \def\bb #1{{\oldbb #1}}
  57
  58 \def\tvi{\vrule height 12pt depth 5pt width 0pt}
  59 \def\tvj{\vrule height 12pt depth 5pt width 1pt}
  60 \def\hfq{\hfill\,\,}
  61 \def\cc#1{\hfq #1\hfq}
  62 \def\tv{\tvi\vrule}
  63 \def\tw{\tvj\vrule}
  64 \def\traithorizontal{\noalign{\hrule}}
  65 \def\traithorizontale{\noalign{\hrule height 1pt}}
  66
  67 \newcommand{\encadre}[1]
  68 {\begin{center}
  69 \fbox{\begin{minipage}{\linewidth}
  70 {#1}
  71 \end{minipage}}
  72 \end{center}
  73 }
  74
  75 \def\pgcd{\mathop{\rm PGCD}\nolimits}
  76 \def\ppcm{\mathop{\rm PPCM}\nolimits}
  77
  78 \def\cut{{}\hfill\cr \hfill{}}
  79
  80 \newcommand{\biindice}[3]%
  81 {
  82 \renewcommand{\arraystretch}{0.5}
  83 \begin{array}[t]{c}
  84 #1\\
  85 {\scriptstyle #2}\\
  86 {\scriptstyle #3}
  87 \end{array}
  88 \renewcommand{\arraystretch}{1}
  89 }
  90
  91 \newlength{\ltxt}
  92 \newcommand{\compo}[4]{
  93 \setlength{\ltxt}{\linewidth}
  94 \setbox#1=\hbox{\includegraphics[scale=#3]{#2.#1}}
  95 \addtolength{\ltxt}{-\wd#1}
  96 \addtolength{\ltxt}{-10pt}
  97 \begin{minipage}{\wd#1}
  98 \includegraphics[scale=#3]{#2.#1}
  99 \end{minipage}
 100 \hfill
 101 \begin{minipage}{\ltxt}
 102 #4
 103 \end{minipage}
 104 }
 105
 106 \newcommand{\compog}[4]{
 107 \setlength{\ltxt}{\linewidth}
 108 \setbox#1=\hbox{\includegraphics[scale=#3]{#2.#1}}
 109 \addtolength{\ltxt}{-\wd#1}
 110 \addtolength{\ltxt}{-10pt}
 111 \begin{minipage}{\ltxt}
 112 #4
 113 \end{minipage}
 114 \hfill
 115 \begin{minipage}{\wd#1}
 116 \includegraphics[scale=#3]{#2.#1}
 117 \end{minipage}
 118 }
 119
 120 \newlength{\lntxt}
 121 \newcommand{\Compo}[4]{
 122 \setlength{\lntxt}{\linewidth}
 123 \setbox#1=\hbox{\includegraphics[scale=#3]{#2}}
 124 \addtolength{\lntxt}{-\wd#1}
 125 \addtolength{\lntxt}{-10pt}
 126 \begin{minipage}{\wd#1}
 127 \includegraphics[scale=#3]{#2}
 128 \end{minipage}
 129 \hfill
 130 \begin{minipage}{\lntxt}
 131 #4
 132 \end{minipage}
 133 }
 134
 135 \newlength{\lnttxt}
 136 \newcommand{\dispo}[3]{
 137 \setlength{\lnttxt}{\linewidth}
 138 \setbox#1=\hbox{#2}
 139 \addtolength{\lnttxt}{-\wd#1}
 140 \addtolength{\lnttxt}{-20pt}
 141 \begin{minipage}{\wd#1}
 142 #2
 143 \end{minipage}
 144 \hfill
 145 \begin{minipage}{\lnttxt}
 146 #3
 147 \end{minipage}
 148 }
 149
 150 \newcounter{num}[section]
 151 \newcommand{\exo}{\addtocounter{num}{1}
 152 \par\underline{\bf Exercice~\thenum} }
 153
 154 \newcommand{\titrage}[2]{
 155 {\Large #1}\hfill#2
 156 \par\rule[+6pt]{\linewidth}{0.5mm}
 157 \par
 158 }
 159
 160 \newcommand{\titragedossier}[1]{
 161 {\small #1}\hfill{\small www.melusine.eu.org/syracuse/poulecl/}
 162 \par\rule[+6pt]{\linewidth}{0.5mm}
 163 \par
 164 }
 165
 166 \newcommand{\partie}[2]{
 167 \begin{center}
 168 \begin{minipage}{#1pt}
 169 \begin{center}
 170 \boxput*(0,0){\colorbox{white}{#2}}
 171 {\rule{\linewidth}{0.5mm}}
 172 \end{center}
 173 \end{minipage}
 174 \end{center}
 175 \par
 176 }
 177
 178 \newenvironment{myenumerate}{
 179 \renewcommand{\theenumi}{\arabic{enumi}}
 180 \def\labelenumi{{\bf \theenumi /}}
 181 \begin{enumerate}}{\end{enumerate}}
 182
 183 \newenvironment{Myenumerate}{
 184 \renewcommand{\theenumi}{\arabic{enumi}}
 185 \def\labelenumi{$\rhd${\bf \theenumi /}}
 186 \begin{enumerate}}{\end{enumerate}}
 187
 188 \newdimen\shadeshift\shadeshift=1pt
 189 \def\shadedtext#1{{\setbox0=\hbox{#1}\leavevmode
 190         \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}%
 191         \kern0.1em\lower0.1em\copy0
 192         \special{color pop}}\vss}\box0}}%
 193 \long\def\shadedparagraph#1\par{{\setbox0=\vbox{\hsize=\hsize#1}%
 194         \noindent\leavevmode
 195         \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}%
 196         \kern0.1em\lower0.1em\copy0
 197         \special{color pop}}\vss}\box0\par}}%
 198
 199 \newboolean{exact}
 200 \setboolean{exact}{true}
 201 \newboolean{racine}
 202 \setboolean{racine}{false}
 203
 204 \newcommand{\pythahypo}[5]{
 205 \opset{decimalsepsymbol={,}}
 206 \opcopy{#4}{A1}
 207 \opcopy{#5}{A2}
 208 Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire :
 209 \[\Eqalign{
 210 #1#3^2&=#1#2^2+#2#3^2\cr
 211 #1#3^2&=\opprint{A1}^2+\opprint{A2}^2\cr
 212 #1#3^2&=\opmul*{A1}{A1}{a1}\opprint{a1}+\opmul*{A2}{A2}{a2}\opprint{a2}\cr
 213 #1#3^2&=\opadd*{a1}{a2}{a3}\opprint{a3}\cr
 214 #1#3&=\sqrt{\opprint{a3}}\cr
 215 \ifthenelse{\boolean{racine}}{}{\ifthenelse{\boolean{exact}}{#1#3&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#3&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4}
 216 \opprint{a4}}}\cr
 217 }\]
 218 }
 219
 220 \newcommand{\pythadroit}[5]{
 221 \opset{decimalsepsymbol={,}}
 222 \opcopy{#4}{A1}
 223 \opcopy{#5}{A2}
 224 Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire :
 225 \[\Eqalign{
 226 #1#3^2&=#1#2^2+#2#3^2\cr
 227 \opprint{A1}^2&=#1#2^2+\opprint{A2}^2\cr
 228 \opmul*{A1}{A1}{a1}\opprint{a1}&=#1#2^2+\opmul*{A2}{A2}{a2}\opprint{a2}\cr
 229 #1#2^2&=\opmul*{A1}{A1}{a1}\opprint{a1}-\opmul*{A2}{A2}{a2}\opprint{a2}\cr
 230 #1#2^2&=\opsub*{a1}{a2}{a3}\opprint{a3}\cr
 231 #1#2&=\sqrt{\opprint{a3}}\cr
 232 \ifthenelse{\boolean{racine}}{}{\ifthenelse{\boolean{exact}}{#1#2&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#2&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4}
 233 \opprint{a4}}}\cr
 234 }\]
 235 }
 236
 237 \newcommand{\Recipytha}[6]{
 238 \opset{decimalsepsymbol={,}}
 239 \opcopy{#4}{A1}
 240 \opcopy{#5}{A2}
 241 \opcopy{#6}{A3}
 242 Dans le triangle $#1#2#3$, $[#1#3]$ est le plus grand côté.
 243 \[\left.
 244   \begin{array}{l}
 245     #1#3^2=\opprint{A1}^2=\opmul*{A1}{A1}{a1}\opprint{a1}\cr
 246     #1#2^2+#2#3^2=\opprint{A2}^2+\opprint{A3}^2=\opmul*{A2}{A2}{a2}\opprint{a2}+\opmul*{A3}{A3}{a3}\opprint{a3}=\opadd*{a2}{a3}{a4}\opprint{a4}\cr
 247   \end{array}
 248 \right\}#1#3^2=#1#2^2+#2#3^2
 249 \]
 250 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.}
 251
 252 \newcommand{\Recipythacol}[6]{
 253 \opset{decimalsepsymbol={,}}
 254 \opcopy{#4}{A1}
 255 \opcopy{#5}{A2}
 256 \opcopy{#6}{A3}
 257 Dans le triangle $#1#2#3$, $[#1#3]$ est le plus grand côté.
 258 \[\Eqalign{
 259     #1#3^2&\kern0.15\linewidth&#1#2^2+#2#3^2\cr
 260     \opprint{A1}^2&&\opprint{A2}^2+\opprint{A3}^2\cr
 261     \opmul*{A1}{A1}{a1}\opprint{a1}&&\opmul*{A2}{A2}{a2}\opprint{a2}+\opmul*{A3}{A3}{a3}\opprint{a3}\cr
 262     &&\opadd*{a2}{a3}{a4}\opprint{a4}\cr
 263 }\]
 264 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.}
 265
 266 \newcommand{\Thales}[5]{%
 267 Dans le triangle $#1#2#3$, $#4$ est un point de la
 268  droite $(#1#2)$, $#5$ est un point de la droite
 269  $(#1#3)$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles.
 270  Le théorème de Thalès permet d'écrire :
 271 \[\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}\]%
 272 }
 273
 274 \newcommand{\Thalesf}[5]{
 275 Dans le triangle $#1#2#3$, $#4$ est un point du
 276  segment $[#1#2]$, $#5$ est un point du segment
 277  $[#1#3]$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles.
 278  L'égalité des 3 rapports permet d'écrire :
 279 \[\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}\]
 280 }
 281
 282 \newcommand{\ResolThales}[6]{%
 283 \opset{decimalsepsymbol={,}}%
 284 \opcopy{#3}{a3}%
 285 \opcopy{#4}{a4}%
 286 \opcopy{#5}{a5}%
 287 On utilise %
 288 \[\Eqalign{%
 289 \frac{#1#2}{\opprint{a3}}&=\frac{\opprint{a4}}{\opprint{a5}}\cr%
 290 #1#2&=\frac{\opprint{a3}\times\opprint{a4}}{\opprint{a5}}\cr%
 291 #1#2&=\frac{\opmul*{a3}{a4}{a6}\opunzero{a6}\opprint{a6}}{\opprint{a5}}\cr%
 292 \ifthenelse{\boolean{exact}}{#1#2&=\opdiv*[maxdivstep=3]{a6}{a5}{a7}{a8}\opunzero{a7}\opprint{a7}\cr}{#1#2&\approx\opdiv*[maxdivstep=3]{a6}{a5}{a7}{a8}\opunzero{a7}\opprint{a7}\cr}%
 293 }\]%
 294 \ifthenelse{\boolean{exact}}{La longueur $#1#2$ mesure \opprint{a7}\,#6}{La longueur $#1#2$ mesure environ \opprint{a7}\,#6}%
 295 }
 296
 297 %%QCM
 298 \newcounter{qqcm}
 299
 300 %définir un booléen qui permet de choisir la correction ou non
 301 \newboolean{solution}
 302
 303 %définir une commande \V qui permet de changer le carré en carré coché suivant la valeur du booléen.
 304 \newcommand{\V}[1]{\ifthenelse{\boolean{solution}}{$\boxtimes$\kern2mm #1}{$\Box$\kern2mm #1}}
 305 \newcommand{\F}[1]{$\Box$\kern2mm #1}
 306 \newcommand{\vr}{\ifthenelse{\boolean{solution}}{$\boxtimes$}{$\Box$}}
 307 \newcommand{\fa}{$\Box$}
 308
 309 %%QCM Version 2
 310 \newenvironment{Qcm}[1][2]{\par\setboolean{solution}{false}
 311 \setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
 312 \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular}
 313 \renewcommand{\arraystretch}{1}}
 314
 315 \newenvironment{Qcmcor}[1][2]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
 316 \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular}
 317 \renewcommand{\arraystretch}{1}}
 318
 319 \newcounter{taill}
 320 \newcommand{\QCM}[3]{\setboolean{solution}{false}
 321 \setcounter{qqcm}{0}
 322 \renewcommand{\arraystretch}{1.5}
 323 \setcounter{taill}{#1}
 324 \addtocounter{taill}{1}
 325 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 326 \hline
 327 \multicolumn{\thetaill}{|c|}{{\sc #2}}\\
 328 #3
 329 \hline
 330 \end{tabularx}
 331 \renewcommand{\arraystretch}{1}
 332 }
 333
 334 \newcommand{\QCMcor}[3]{\setboolean{solution}{true}
 335 \setcounter{qqcm}{0}
 336 \renewcommand{\arraystretch}{1.5}
 337 \setcounter{taill}{#1}
 338 \addtocounter{taill}{1}
 339 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 340 \hline
 341 \multicolumn{\thetaill}{|c|}{{\sc #2}}\\
 342 #3
 343 \hline
 344 \end{tabularx}
 345 \renewcommand{\arraystretch}{1}
 346 }
 347
 348 \newcommand{\QCMvar}[4]{\setboolean{solution}{false}
 349 \setcounter{qqcm}{0}
 350 \renewcommand{\arraystretch}{#2}
 351 \setcounter{taill}{#1}
 352 \addtocounter{taill}{1}
 353 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 354 \hline
 355 \multicolumn{\thetaill}{|c|}{{\sc #3}}\\
 356 #4
 357 \hline
 358 \end{tabularx}
 359 \renewcommand{\arraystretch}{1}
 360 }
 361
 362 \newcommand{\QCMvarcor}[4]{\setboolean{solution}{true}
 363 \setcounter{qqcm}{0}
 364 \renewcommand{\arraystretch}{#2}
 365 \setcounter{taill}{#1}
 366 \addtocounter{taill}{1}
 367 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 368 \hline
 369 \multicolumn{\thetaill}{|c|}{{\sc #3}}\\
 370 #4
 371 \hline
 372 \end{tabularx}
 373 \renewcommand{\arraystretch}{1}
 374 }
 375
 376 \newcommand{\QCMsimple}[2]{\setboolean{solution}{false}
 377 \setcounter{qqcm}{0}
 378 \renewcommand{\arraystretch}{1.5}
 379 \setcounter{taill}{#1}
 380 \addtocounter{taill}{1}
 381 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 382 \hline
 383 #2
 384 \hline
 385 \end{tabularx}
 386 \renewcommand{\arraystretch}{1}
 387 }
 388
 389 \newcommand{\QCMsimplevar}[3]{\setboolean{solution}{false}
 390 \setcounter{qqcm}{0}
 391 \renewcommand{\arraystretch}{#2}
 392 \setcounter{taill}{#1}
 393 \addtocounter{taill}{1}
 394 \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}}
 395 \hline
 396 #3
 397 \hline
 398 \end{tabularx}
 399 \renewcommand{\arraystretch}{1}
 400 }
 401
 402 \newenvironment{VF}[1]{\par\setboolean{solution}{false}
 403 \setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
 404 \begin{center}
 405 \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline
 406 \multicolumn{3}{|c|}{#1}\\
 407 \hline
 408 \multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\
 409 \hline
 410 }{\hline\end{tabular}
 411 \end{center}\renewcommand{\arraystretch}{1}}
 412
 413 \newenvironment{VFcor}[1]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5}
 414 \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline
 415 \multicolumn{3}{|c|}{#1}\\
 416 \hline
 417 \multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\
 418 \hline
 419 }{\hline\end{tabular}
 420 \renewcommand{\arraystretch}{1}}