1 %=========================================
3 %christophe.poulain@melusine.eu.org
4 %création : 25 Septembre 1999
5 %dernière modification : 28 Avril 2005
6 %=========================================
8 \RequirePackage{xlop,fancybox,
color,amssymb,ifthen
}
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
}
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
} }
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}
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}}\,
}
44 \newcommand{\vecteur}[1]
45 {\overrightarrow{\strut #1}}
51 \textfont\bbfam=
\tenbb
52 \scriptfont\bbfam=
\sevenbb
53 \scriptscriptfont\bbfam=
\fivebb
54 \def\bb{\fam\bbfam\tenbb}
56 \def\bb #1{{\oldbb #1}}
58 \def\tvi{\vrule height
12pt depth
5pt width
0pt
}
59 \def\tvj{\vrule height
12pt depth
5pt width
1pt
}
61 \def\cc#1{\hfq #1\hfq}
64 \def\traithorizontal{\noalign{\hrule}}
65 \def\traithorizontale{\noalign{\hrule height
1pt
}}
67 \newcommand{\encadre}[1]
69 \fbox{\begin{minipage
}{\linewidth}
75 \def\pgcd{\mathop{\rm PGCD
}\nolimits}
76 \def\ppcm{\mathop{\rm PPCM
}\nolimits}
78 \def\cut{{}\hfill\cr \hfill{}}
80 \newcommand{\biindice}[3]%
82 \renewcommand{\arraystretch}{0.5}
88 \renewcommand{\arraystretch}{1}
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}
101 \begin{minipage
}{\ltxt}
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}
115 \begin{minipage
}{\wd#1}
116 \includegraphics[scale=
#3]{#2.
#1}
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}
130 \begin{minipage
}{\lntxt}
136 \newcommand{\dispo}[3]{
137 \setlength{\lnttxt}{\linewidth}
139 \addtolength{\lnttxt}{-
\wd#1}
140 \addtolength{\lnttxt}{-
20pt
}
141 \begin{minipage
}{\wd#1}
145 \begin{minipage
}{\lnttxt}
150 \newcounter{num
}[section
]
151 \newcommand{\exo}{\addtocounter{num
}{1}
152 \par\underline{\bf Exercice~
\thenum} }
154 \newcommand{\titrage}[2]{
156 \par\rule[+
6pt
]{\linewidth}{0.5mm
}
160 \newcommand{\titragedossier}[1]{
161 {\small #1}\hfill{\small www.melusine.eu.org/syracuse/poulecl/
}
162 \par\rule[+
6pt
]{\linewidth}{0.5mm
}
166 \newcommand{\partie}[2]{
168 \begin{minipage
}{#1pt
}
170 \boxput*(
0,
0)
{\colorbox{white
}{#2}}
171 {\rule{\linewidth}{0.5mm
}}
178 \newenvironment{myenumerate
}{
179 \renewcommand{\theenumi}{\arabic{enumi
}}
180 \def\labelenumi{{\bf \theenumi /
}}
181 \begin{enumerate
}}{\end{enumerate
}}
183 \newenvironment{Myenumerate
}{
184 \renewcommand{\theenumi}{\arabic{enumi
}}
185 \def\labelenumi{$
\rhd$
{\bf \theenumi /
}}
186 \begin{enumerate
}}{\end{enumerate
}}
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}%
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}}%
200 \setboolean{exact
}{true
}
202 \setboolean{racine
}{false
}
204 \newcommand{\pythahypo}[5]{
205 \opset{decimalsepsymbol=
{,
}}
208 Dans le triangle $
#1#2#3$ rectangle en $
#2$, le théorème de Pythagore permet d'écrire :
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
}
220 \newcommand{\pythadroit}[5]{
221 \opset{decimalsepsymbol=
{,
}}
224 Dans le triangle $
#1#2#3$ rectangle en $
#2$, le théorème de Pythagore permet d'écrire :
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
}
237 \newcommand{\Recipytha}[6]{
238 \opset{decimalsepsymbol=
{,
}}
242 Dans le triangle $
#1#2#3$, $
[#1#3]$ est le plus grand côté.
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
248 \right\
}#1#3^
2=
#1#2^
2+
#2#3^
2
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.
}
252 \newcommand{\Recipythacol}[6]{
253 \opset{decimalsepsymbol=
{,
}}
257 Dans le triangle $
#1#2#3$, $
[#1#3]$ est le plus grand côté.
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
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.
}
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}\
]%
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}\
]
282 \newcommand{\ResolThales}[6]{%
283 \opset{decimalsepsymbol=
{,
}}%
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}%
294 \ifthenelse{\boolean{exact
}}{La longueur $
#1#2$ mesure
\opprint{a7
}\,
#6}{La longueur $
#1#2$ mesure environ
\opprint{a7
}\,
#6}%
300 %définir un booléen qui permet de choisir la correction ou non
301 \newboolean{solution
}
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$
}
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}}
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}}
320 \newcommand{\QCM}[3]{\setboolean{solution
}{false
}
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|
}}
327 \multicolumn{\thetaill}{|c|
}{{\sc #2}}\\
331 \renewcommand{\arraystretch}{1}
334 \newcommand{\QCMcor}[3]{\setboolean{solution
}{true
}
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|
}}
341 \multicolumn{\thetaill}{|c|
}{{\sc #2}}\\
345 \renewcommand{\arraystretch}{1}
348 \newcommand{\QCMvar}[4]{\setboolean{solution
}{false
}
350 \renewcommand{\arraystretch}{#2}
351 \setcounter{taill
}{#1}
352 \addtocounter{taill
}{1}
353 \begin{tabularx
}{\linewidth}{|>
{\small\stepcounter{qqcm
}{\bf \theqqcm/
}\,
}X|*
{#1}{l|
}}
355 \multicolumn{\thetaill}{|c|
}{{\sc #3}}\\
359 \renewcommand{\arraystretch}{1}
362 \newcommand{\QCMvarcor}[4]{\setboolean{solution
}{true
}
364 \renewcommand{\arraystretch}{#2}
365 \setcounter{taill
}{#1}
366 \addtocounter{taill
}{1}
367 \begin{tabularx
}{\linewidth}{|>
{\small\stepcounter{qqcm
}{\bf \theqqcm/
}\,
}X|*
{#1}{l|
}}
369 \multicolumn{\thetaill}{|c|
}{{\sc #3}}\\
373 \renewcommand{\arraystretch}{1}
376 \newcommand{\QCMsimple}[2]{\setboolean{solution
}{false
}
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|
}}
386 \renewcommand{\arraystretch}{1}
389 \newcommand{\QCMsimplevar}[3]{\setboolean{solution
}{false
}
391 \renewcommand{\arraystretch}{#2}
392 \setcounter{taill
}{#1}
393 \addtocounter{taill
}{1}
394 \begin{tabularx
}{\linewidth}{|>
{\small\stepcounter{qqcm
}{\bf \theqqcm/
}\,
}X|*
{#1}{l|
}}
399 \renewcommand{\arraystretch}{1}
402 \newenvironment{VF
}[1]{\par\setboolean{solution
}{false
}
403 \setcounter{qqcm
}{0}\renewcommand{\arraystretch}{1.5}
405 \begin{tabular
}{|>
{\small\stepcounter{qqcm
}{\bf \theqqcm/
}\,
}b
{\linewidth/
2}|*
{2}{c|
}}\hline
406 \multicolumn{3}{|c|
}{#1}\\
408 \multicolumn{1}{|c|
}{\bf Question
}&
\multicolumn{1}{c|
}{\bf Vrai
}&
\multicolumn{1}{c|
}{\bf Faux
}\\
410 }{\hline\end{tabular
}
411 \end{center
}\renewcommand{\arraystretch}{1}}
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}\\
417 \multicolumn{1}{|c|
}{\bf Question
}&
\multicolumn{1}{c|
}{\bf Vrai
}&
\multicolumn{1}{c|
}{\bf Faux
}\\
419 }{\hline\end{tabular
}
420 \renewcommand{\arraystretch}{1}}