%%% % Le th\'eor\`eme de Pythagore %%% \setKVdefault[ClesPythagore]{Exact=false,AvantRacine=false,Racine=false,Entier=false,Egalite=false,Precision=2,Soustraction=false,Figure=false,FigureSeule=false,Angle=0,Echelle=1cm,Reciproque=false,ReciColonnes=false,Faible=false,Unite=cm,EnchaineA=false,EnchaineB=false,EnchaineC=false,Perso=false,AllPerso=false,SansMots=false} \defKV[ClesPythagore]{ValeurA=\setKV[ClesPythagore]{EnchaineA}} \defKV[ClesPythagore]{ValeurB=\setKV[ClesPythagore]{EnchaineB}} \defKV[ClesPythagore]{ValeurC=\setKV[ClesPythagore]{EnchaineC}} % On d\'efinit les figures \`a utiliser \def\MPFigurePytha#1#2#3#4#5#6{% % #1 Premier sommet % #2 Sommet de l'angle droit % #3 troisi\`eme sommet % #4 1ere longueur % #5 2eme longueur % #6 angle de rotation de la figure \ifluatex \mplibforcehmode \begin{mplibcode} u:=\useKV[ClesPythagore]{Echelle}; pair A,B,C,O,D,E,F;%B est le sommet de l'angle droit O=u*(2.5,2.5); path cc; cc=(fullcircle scaled 4u) shifted O; % On place les points A,B,C sur le cercle de mani\`ere \`a faciliter la rotation de la figure A=point(0.9*length cc) of cc; B=A rotatedabout(O,-120); C=2[A,O]; % On tourne pour \'eventuellement moins de lassitude :) A:=A rotatedabout(O,#6); B:=B rotatedabout(O,#6); C:=C rotatedabout(O,#6); % On d\'efinit l'angle droit D-B=7*unitvector(C-B); F-B=7*unitvector(A-B); E-D=F-B; draw A{dir(angle(B-A)+5)}..B{dir(angle(B-A)+5)}; draw B{dir(angle(C-B)+5)}..C{dir(angle(C-B)+5)}; draw C{dir(angle(A-C)+5)}..A{dir(angle(A-C)+5)}; draw D--E--F; numeric decalage; decalage=3mm; if (#4<#5) or (#4=#5) : if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,B]-decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); else: label(btex \num{#4} etex,1/2[C,B]-decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); fi else: if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,A]-decalage*(unitvector(C-A) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); else: label(btex \num{#4} etex,1/2[A,C]+decalage*(unitvector(A-C) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); fi; fi; label(btex #3 etex,1.2[O,A]); label(btex #2 etex,1.2[O,B]); label(btex #1 etex,1.2[O,C]); \end{mplibcode} \else \begin{mpost}[mpsettings={u:=\useKV[ClesPythagore]{Echelle};}] pair A,B,C,O,D,E,F;%B est le sommet de l'angle droit O=u*(2.5,2.5); path cc; cc=(fullcircle scaled 4u) shifted O; % On place les points A,B,C sur le cercle de mani\`ere \`a faciliter la rotation de la figure A=point(0.9*length cc) of cc; B=A rotatedabout(O,-120); C=2[A,O]; % On tourne pour \'eventuellement moins de lassitude :) A:=A rotatedabout(O,#6); B:=B rotatedabout(O,#6); C:=C rotatedabout(O,#6); % On d\'efinit l'angle droit D-B=7*unitvector(C-B); F-B=7*unitvector(A-B); E-D=F-B; draw A{dir(angle(B-A)+5)}..B{dir(angle(B-A)+5)}; draw B{dir(angle(C-B)+5)}..C{dir(angle(C-B)+5)}; draw C{dir(angle(A-C)+5)}..A{dir(angle(A-C)+5)}; draw D--E--F; decalage=3mm; if (#4<#5) or (#4=#5) : if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,B]-decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); else: label(btex \num{#4} etex,1/2[C,B]-decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); fi else: if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,A]-decalage*(unitvector(C-A) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); else: label(btex \num{#4} etex,1/2[A,C]+decalage*(unitvector(A-C) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); fi; fi; label(btex #3 etex,1.2[O,A]); label(btex #2 etex,1.2[O,B]); label(btex #1 etex,1.2[O,C]); \end{mpost} \fi } \def\MPFigureReciPytha#1#2#3#4#5#6#7{% % #1 Premier sommet % #2 Sommet de l'angle droit % #3 troisi\`eme sommet % #4 1ere longueur % #5 2eme longueur % #6 3eme longueur % #7 angle de rotation de la figure \ifluatex \mplibforcehmode \begin{mplibcode} u:=\useKV[ClesPythagore]{Echelle}; pair A,B,C,O,D,E,F;%B est le sommet de l'angle droit O=u*(2.5,2.5); path cc; cc=(fullcircle scaled 4u) shifted O; % On place les points A,B,C sur le cercle de mani\`ere \`a faciliter la rotation de la figure A=point(0.8*length cc) of cc; B=A rotatedabout(O,-100); C=2[A,O]; % On tourne pour \'eventuellement moins de lassitude :) A:=A rotatedabout(O,#7); B:=B rotatedabout(O,#7); C:=C rotatedabout(O,#7); draw A{dir(angle(B-A)+5)}..B{dir(angle(B-A)+5)}; draw B{dir(angle(C-B)+5)}..C{dir(angle(C-B)+5)}; draw C{dir(angle(A-C)+5)}..A{dir(angle(A-C)+5)}; decalage=3mm; if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,A]-decalage*(unitvector(C-A) rotated 90)); label(btex \num{#5} etex,1/2[C,B]-decalage*(unitvector(C-B))); label(btex \num{#6} etex,1/2[A,B]-decalage*(unitvector(C-B))); else: label(btex \num{#4} etex,1/2[A,C]+decalage*(unitvector(A-C) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); label(btex \num{#6} etex,1/2[C,B]-decalage*(unitvector(A-B))); fi; label(btex #1 etex,1.2[O,A]); label(btex #2 etex,1.2[O,B]); label(btex #3 etex,1.2[O,C]); \end{mplibcode} \else \begin{mpost}[mpsettings={u:=\useKV[ClesPythagore]{Echelle};}] pair A,B,C,O,D,E,F;%B est le sommet de l'angle droit O=u*(2.5,2.5); path cc; cc=(fullcircle scaled 4u) shifted O; % On place les points A,B,C sur le cercle de mani\`ere \`a faciliter la rotation de la figure A=point(0.8*length cc) of cc; B=A rotatedabout(O,-100); C=2[A,O]; % On tourne pour \'eventuellement moins de lassitude :) A:=A rotatedabout(O,#7); B:=B rotatedabout(O,#7); C:=C rotatedabout(O,#7); draw A{dir(angle(B-A)+5)}..B{dir(angle(B-A)+5)}; draw B{dir(angle(C-B)+5)}..C{dir(angle(C-B)+5)}; draw C{dir(angle(A-C)+5)}..A{dir(angle(A-C)+5)}; decalage=3mm; if ypart(B)>ypart(O) : label(LATEX("\num{"&decimal(#4)&"}") rotated angle(C-A),1/2[C,A]-decalage*(unitvector(C-A) rotated 90)); label(LATEX("\num{"&decimal(#5)&"}") rotated(angle(C-B)),1/2[C,B]-decalage*(unitvector(C-B))); label(LATEX("\num{"&decimal(#6)&"}") rotated(angle(B-A)),1/2[A,B]-decalage*(unitvector(C-B))); else: label(LATEX("\num{"&decimal(#4)&"}") rotated angle(A-C),1/2[A,C]+decalage*(unitvector(A-C) rotated 90)); label(LATEX("\num{"&decimal(#5)&"}") rotated(angle(A-B)),1/2[A,B]-decalage*(unitvector(C-B))); label(LATEX("\num{"&decimal(#6)&"}") rotated angle(C-B),1/2[C,B]-decalage*(unitvector(A-B))); fi; label(btex #1 etex,1.2[O,A]); label(btex #2 etex,1.2[O,B]); label(btex #3 etex,1.2[O,C]); \end{mpost} \fi } \def\MPFigurePythaSansMots#1#2#3#4#5#6{% % #1 Premier sommet % #2 Sommet de l'angle droit % #3 troisi\`eme sommet % #4 1ere longueur % #5 2eme longueur % #6 angle de rotation de la figure \ifluatex \mplibforcehmode \begin{mplibcode} u:=\useKV[ClesPythagore]{Echelle}; pair A,B,C,O,D,E,F,L[],M[],N[];%B est le sommet de l'angle droit O=u*(2.5,2.5); path cc; cc=(fullcircle scaled 4u) shifted O; % On place les points A,B,C sur le cercle de mani\`ere \`a faciliter la rotation de la figure A=point(0.9*length cc) of cc; B=A rotatedabout(O,-120); C=2[A,O]; % On tourne pour \'eventuellement moins de lassitude :) A:=A rotatedabout(O,#6); B:=B rotatedabout(O,#6); C:=C rotatedabout(O,#6); % On d\'efinit l'angle droit D-B=7*unitvector(C-B); F-B=7*unitvector(A-B); E-D=F-B; trace polygone(A,B,C); draw D--E--F; L1=rotation(B,A,90); L2-L1=B-A; trace chemin(A,L1,L2,B); M1=rotation(C,B,90); M2-M1=C-B; trace chemin(B,M1,M2,C); N1=rotation(A,C,90); N2-N1=A-C; trace chemin(C,N1,N2,A); numeric decalage; decalage=3mm; if (#4<#5) or (#4=#5) : if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,B]+decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]+decalage*(unitvector(C-B))); label(TEX("\begin{tabular}{c}Aire\\$\num{#5}\times\num{#5}$\\$\num{\fpeval{#5*#5}}$\\\end{tabular}"),iso(A,L2)); label(TEX("\begin{tabular}{c}Aire\\$\num{#4}\times\num{#4}$\\$\num{\fpeval{#4*#4}}$\\\end{tabular}"),iso(B,M2)); label(TEX("\begin{tabular}{c}Aire\\$\num{\fpeval{#4*#4}}+\num{\fpeval{#5*#5}}$\\$\num{\fpeval{#4*#4+#5*#5}}$\\\end{tabular}"),iso(C,N2)); else: label(btex \num{#4} etex,1/2[C,B]+decalage*(unitvector(A-B))); label(btex \num{#5} etex,1/2[A,B]+decalage*(unitvector(C-B))); label(TEX("\begin{tabular}{c}Aire\\$\num{#5}\times\num{#5}$\\$\num{\fpeval{#5*#5}}$\\\end{tabular}"),iso(A,L2)); label(TEX("\begin{tabular}{c}Aire\\$\num{#4}\times\num{#4}$\\$\num{\fpeval{#4*#4}}$\\\end{tabular}"),iso(B,M2)); label(TEX("\begin{tabular}{c}Aire\\$\num{\fpeval{#4*#4}}+\num{\fpeval{#5*#5}}$\\$\num{\fpeval{#4*#4+#5*#5}}$\\\end{tabular}"),iso(C,N2)); drawarrow 2/10[iso(A,L2),iso(C,N2)]--8/10[iso(A,L2),iso(C,N2)]; drawarrow 2/10[iso(B,M2),iso(C,N2)]--8/10[iso(B,M2),iso(C,N2)]; fi else: if ypart(B)>ypart(O) : label(btex \num{#4} etex,1/2[C,A]-decalage*(unitvector(C-A) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); label(TEX("\begin{tabular}{c}Aire\\$\num{#5}\times\num{#5}$\\$\num{\fpeval{#5*#5}}$\\\end{tabular}"),iso(A,L2)); label(TEX("\begin{tabular}{c}Aire\\$\num{#4}\times\num{#4}$\\$\num{\fpeval{#4*#4}}$\\\end{tabular}"),iso(C,N2)); label(TEX("\begin{tabular}{c}Aire\\$\num{\fpeval{#4*#4}}-\num{\fpeval{#5*#5}}$\\$\num{\fpeval{#4*#4-#5*#5}}$\\\end{tabular}"),iso(B,M2)); drawarrow 2/10[iso(A,L2),iso(B,M2)]--8/10[iso(A,L2),iso(B,M2)]; drawarrow 2/10[iso(C,N2),iso(B,M2)]--8/10[iso(C,N2),iso(B,M2)]; else: label(btex \num{#4} etex,1/2[A,C]+decalage*(unitvector(A-C) rotated 90)); label(btex \num{#5} etex,1/2[A,B]-decalage*(unitvector(C-B))); label(TEX("\begin{tabular}{c}Aire\\$\num{#5}\times\num{#5}$\\$\num{\fpeval{#5*#5}}$\\\end{tabular}"),iso(A,L2)); label(TEX("\begin{tabular}{c}Aire\\$\num{#4}\times\num{#4}$\\$\num{\fpeval{#4*#4}}$\\\end{tabular}"),iso(C,N2)); label(TEX("\begin{tabular}{c}Aire\\$\num{\fpeval{#4*#4}}-\num{\fpeval{#5*#5}}$\\$\num{\fpeval{#4*#4-#5*#5}}$\\\end{tabular}"),iso(B,M2)); drawarrow 2/10[iso(A,L2),iso(B,M2)]--8/10[iso(A,L2),iso(B,M2)]; drawarrow 2/10[iso(C,N2),iso(B,M2)]--8/10[iso(C,N2),iso(B,M2)]; fi; fi; label(btex #3 etex,1.2[O,A]); label(btex #2 etex,1.2[O,B]); label(btex #1 etex,1.2[O,C]); \end{mplibcode} \fi } \newcommand\RedactionPythagore{}% \newcommand\RedactionReciPythagore{}% \newcommand\RedactionCalculsPythagore{}% \newcommand\RedactionCalculsReciPythagore{}% \newcommand\RedactionConclusionReciPythagore{}% \NewDocumentCommand\CalculsPythagore{o m m m m}{ \xintifboolexpr{#3<#4 || #3==#4}{%\ifnum#3<#4% \xdef\ResultatPytha{\fpeval{round(sqrt(#3^2+#4^2),\useKV[ClesPythagore]{Precision})}}% \begin{align*} \NomA\NomC^2&=\NomA\NomB^2+\NomB\NomC^2\\ \NomA\NomC^2&=\ifboolKV[ClesPythagore]{EnchaineA}{\opcopy{\useKV[ClesPythagore]{ValeurA}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opexport{A1}{\Aun}\num{\Aun}^2}+\ifboolKV[ClesPythagore]{EnchaineB}{\opcopy{\useKV[ClesPythagore]{ValeurB}}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}{\opexport{A2}{\Adeux}\num{\Adeux}^2}\\ \NomA\NomC^2&=\ifboolKV[ClesPythagore]{EnchaineA}{\opexport{a1}{\Aun}\num{\Aun}}{\opmul*{A1}{A1}{a1}\opexport{a1}{\Aun}\num{\Aun}}+\ifboolKV[ClesPythagore]{EnchaineB}{\opexport{a2}{\Adeux}\num{\Adeux}}{\opmul*{A2}{A2}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}\\ \NomA\NomC^2&=\opadd*{a1}{a2}{a3}\opexport{a3}{\Atrois}\num{\Atrois}%\\ \ifboolKV[ClesPythagore]{AvantRacine}{}{% \ifboolKV[ClesPythagore]{Entier}{}{\\\NomA\NomC&=\sqrt{\opexport{a3}{\Atrois}\num{\Atrois}}} \ifboolKV[ClesPythagore]{Racine}{}{\\\NomA\NomC&\IfInteger{\fpeval{round(sqrt(#3^2+#4^2),\useKV[ClesPythagore]{Precision})^2-#3^2-#4^2}}{=}{\approx}\opsqrt[maxdivstep=5]{a3}{a4}\opround{a4}{pres}{a4}\opunzero{a4}\opexport{a4}{\Aquatre}\SI{\Aquatre}{\PfCPythaUnit}}%\\ } \end{align*} }{%\else \xdef\ResultatPytha{\fpeval{round(sqrt(#3^2-#4^2),\useKV[ClesPythagore]{Precision})}}% \ifboolKV[ClesPythagore]{Soustraction}{% \begin{align*} \NomA\NomB^2&=\NomA\NomC^2-\NomB\NomC^2\\ \NomA\NomB^2&=\ifboolKV[ClesPythagore]{EnchaineC}{\opcopy{\useKV[ClesPythagore]{ValeurC}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opexport{A1}{\Aun}\num{\Aun}^2}-\ifboolKV[ClesPythagore]{EnchaineB}{\opcopy{\useKV[ClesPythagore]{ValeurB}}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}{\opexport{A2}{\Adeux}\num{\Adeux}^2}\\ \NomA\NomB^2&=\ifboolKV[ClesPythagore]{EnchaineC}{\opcopy{\useKV[ClesPythagore]{ValeurC}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opmul*{A1}{A1}{a1}\opexport{a1}{\Aun}\num{\Aun}}-\ifboolKV[ClesPythagore]{EnchaineB}{\opexport{a2}{\Adeux}\num{\Adeux}}{\opmul*{A2}{A2}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}\\ \NomA\NomB^2&=\opsub*{a1}{a2}{a3}\opexport{a3}{\Atrois}\num{\Atrois}%\\ \ifboolKV[ClesPythagore]{AvantRacine}{}{% \ifboolKV[ClesPythagore]{Entier}{}{\\\NomA\NomB&=\sqrt{\opexport{a3}{\Atrois}\num{\Atrois}}} \ifboolKV[ClesPythagore]{Racine}{}{\\\NomA\NomB&\IfInteger{\fpeval{round(sqrt(#3^2-#4^2),\useKV[ClesPythagore]{Precision})^2-#3^2+#4^2}}{=}{\approx}\opsqrt[maxdivstep=5]{a3}{a4}\opround{a4}{pres}{a4}\opunzero{a4}\opexport{a4}{\Aquatre}\SI{\Aquatre}{\PfCPythaUnit}}%\\ } \end{align*} }{% \begin{align*} \NomA\NomC^2&=\NomA\NomB^2+\NomB\NomC^2\\ \ifboolKV[ClesPythagore]{EnchaineC}{\opcopy{\useKV[ClesPythagore]{ValeurC}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opexport{A1}{\Aun}\num{\Aun}^2}&=\NomA\NomB^2+\ifboolKV[ClesPythagore]{EnchaineB}{\opcopy{\useKV[ClesPythagore]{ValeurB}}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}{\opexport{A2}{\Adeux}\num{\Adeux}^2}\\ \ifboolKV[ClesPythagore]{EnchaineC}{\opcopy{\useKV[ClesPythagore]{ValeurC}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opmul*{A1}{A1}{a1}\opexport{a1}{\Aun}\num{\Aun}}&=\NomA\NomB^2+\ifboolKV[ClesPythagore]{EnchaineB}{\opexport{a2}{\Adeux}\num{\Adeux}}{\opmul*{A2}{A2}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}\\ \NomA\NomB^2&=\ifboolKV[ClesPythagore]{EnchaineC}{\opcopy{\useKV[ClesPythagore]{ValeurC}}{a1}\opexport{a1}{\Aun}\num{\Aun}}{\opmul*{A1}{A1}{a1}\opexport{a1}{\Aun}\num{\Aun}}-\ifboolKV[ClesPythagore]{EnchaineB}{\opexport{a2}{\Adeux}\num{\Adeux}}{\opmul*{A2}{A2}{a2}\opexport{a2}{\Adeux}\num{\Adeux}}\\ \NomA\NomB^2&=\opsub*{a1}{a2}{a3}\opexport{a3}{\Atrois}\num{\Atrois}%\\ \ifboolKV[ClesPythagore]{AvantRacine}{}{% \ifboolKV[ClesPythagore]{Entier}{}{\\\NomA\NomB&=\sqrt{\opexport{a3}{\Atrois}\num{\Atrois}}}% \ifboolKV[ClesPythagore]{Racine}{}{\\\NomA\NomB&\IfInteger{\fpeval{round(sqrt(#3^2-#4^2),\useKV[ClesPythagore]{Precision})^2-#3^2+#4^2}}{=}{\approx}\opsqrt[maxdivstep=5]{a3}{a4}\opround{a4}{pres}{a4}\opunzero{a4}\opexport{a4}{\Aquatre}\SI{\Aquatre}{\PfCPythaUnit}}%\\ } \end{align*} }% }%\fi% } \NewDocumentCommand\PfCCalculsReciPythagore{ommmm}{% \ifboolKV[ClesPythagore]{ReciColonnes}{% \[ \begin{array}{cccc|cccc} &&\NomA\NomC^2&&&\NomA\NomB^2&+&\NomB\NomC^2\\ &&\ifboolKV[ClesPythagore]{EnchaineC}{% \num{\useKV[ClesPythagore]{ValeurC}}% \xdef\PfCRetiensReciHypo{\useKV[ClesPythagore]{ValeurC}}% }{% \num{#3}^2}&&&\ifboolKV[ClesPythagore]{EnchaineA}{\num{\useKV[ClesPythagore]{ValeurA}}}{\num{#4}^2}&+&\ifboolKV[ClesPythagore]{EnchaineB}{\num{\useKV[ClesPythagore]{ValeurB}}}{\num{#5}^2}\\ &&&&&\ifboolKV[ClesPythagore]{EnchaineA}{%Enchaine A \ifboolKV[ClesPythagore]{EnchaineB}{%EnchaineB }{%Pas Enchaine B \num{\useKV[ClesPythagore]{ValeurA}}}}{%Pas EnchaineA \num{\fpeval{#4*#4}}}&\ifboolKV[ClesPythagore]{EnchaineA}{\ifboolKV[ClesPythagore]{EnchaineB}{}{+}}{+}&\ifboolKV[ClesPythagore]{EnchaineB}{%Enchaine B \ifboolKV[ClesPythagore]{EnchaineA}{%EnchaineA }{%Pas Enchaine A \num{\useKV[ClesPythagore]{ValeurB}}}}{%Pas EnchaineB \num{\fpeval{#5*#5}}}\\ &&\ifboolKV[ClesPythagore]{EnchaineC}{}{% \xdef\PfCRetiensReciHypo{\fpeval{#3*#3}}% \num{\PfCRetiensReciHypo} }&&&\multicolumn{3}{c}{% \ifboolKV[ClesPythagore]{EnchaineA}{%Enchaine A \ifboolKV[ClesPythagore]{EnchaineB}{%EnchaineB \xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurA}+\useKV[ClesPythagore]{ValeurB}}}% }{%Pas Enchaine B \xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurA}+#5*#5}}% }}{%Pas EnchaineA \ifboolKV[ClesPythagore]{EnchaineB}{ \xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurB}+#4*#4}}% }{% \xdef\PfCRetiensReciSomme{\fpeval{#4*#4+#5*#5}}% } } \num{\PfCRetiensReciSomme}% }\\ \end{array} \] }{% \[\left. \begin{array}{l} \NomA\NomC^2=\ifboolKV[ClesPythagore]{EnchaineC}{% \num{\useKV[ClesPythagore]{ValeurC}}% \xdef\PfCRetiensReciHypo{\useKV[ClesPythagore]{ValeurC}}% }{% \num{#3}^2=\num{\fpeval{#3*#3}} \xdef\PfCRetiensReciHypo{\fpeval{#3*#3}}% }\\ \NomA\NomB^2+\NomB\NomC^2=\ifboolKV[ClesPythagore]{EnchaineA}{\num{\useKV[ClesPythagore]{ValeurA}}}{\num{#4}^2}+\ifboolKV[ClesPythagore]{EnchaineB}{\num{\useKV[ClesPythagore]{ValeurB}}}{\num{#5}^2}= \ifboolKV[ClesPythagore]{EnchaineA}{%Enchaine A \ifboolKV[ClesPythagore]{EnchaineB}{%Enchaine B \xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurA}+\useKV[ClesPythagore]{ValeurB}}}% \num{\PfCRetiensReciSomme}% }{% PasEnchaine B \num{\useKV[ClesPythagore]{ValeurA}}+\num{\fpeval{#5*#5}}=% \xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurA}+#5*#5}}% \num{\PfCRetiensReciSomme}% } }{%Pas Enchaine A \num{\fpeval{#4*#4}}+ \ifboolKV[ClesPythagore]{EnchaineB}{%Enchaine B \num{\useKV[ClesPythagore]{ValeurB}}=\xdef\PfCRetiensReciSomme{\fpeval{\useKV[ClesPythagore]{ValeurB}+#4*#4}}% \num{\PfCRetiensReciSomme}% }{%Pas Enchaine B \num{\fpeval{#5*#5}}=\xdef\PfCRetiensReciSomme{\fpeval{#5*#5+#4*#4}}% \num{\PfCRetiensReciSomme}% } } \end{array} \right\}\xintifboolexpr{\PfCRetiensReciHypo==\PfCRetiensReciSomme}{\NomA\NomC^2=\NomA\NomB^2+\NomB\NomC^2}{\NomA\NomC^2\not=\NomA\NomB^2+\NomB\NomC^2}% \] }% }% % \newcommand\Pythagore[5][]{% \NewDocumentCommand\Pythagore{o m m m m}{% % #1 Param\`etres sous forme de cl\'es % #2 Nom "complet" du triangle : ABC par exemple % #3 Premi\`ere longueur % #4 Deuxi\`eme longueur % #5 Troisi\`eme longueur (\'eventuellement vide) \useKVdefault[ClesPythagore]% \setKV[ClesPythagore]{#1}% \DeclareSIUnit{\PfCPythaUnit}{\useKV[ClesPythagore]{Unite}}% \ifboolKV[ClesPythagore]{Reciproque}{% % On retient les noms des sommets \StrMid{#2}{1}{1}[\NomA]% \StrMid{#2}{2}{2}[\NomB]% \StrMid{#2}{3}{3}[\NomC]% \xdef\NomTriangle{\NomA\NomB\NomC}% % on stocke les valeurs donn\'ees \opcopy{#3}{A1}% \opcopy{#4}{A2}% \opcopy{#5}{A3}% \xdef\GrandCote{#3}% \xdef\PetitCote{#4}% \xdef\MoyenCote{#5}% % On trace une figure ou pas ? \ifboolKV[ClesPythagore]{FigureSeule}{% \MPFigureReciPytha{\NomA}{\NomB}{\NomC}{#3}{#4}{#5}{\useKV[ClesPythagore]{Angle}}% }{% \ifboolKV[ClesPythagore]{Figure}{%Utilisation obligatoire de l'option --shell-escape de la compilation \begin{multicols}{2} {\em La figure est donn\'ee \`a titre indicatif.}% \[\MPFigureReciPytha{\NomA}{\NomB}{\NomC}{#3}{#4}{#5}{\useKV[ClesPythagore]{Angle}}\]% \par\columnbreak\par% \ifboolKV[ClesPythagore]{AllPerso}{% \RedactionReciPythagore% \RedactionCalculsReciPythagore% \RedactionConclusionReciPythagore% }{% % on r\'edige \ifboolKV[ClesPythagore]{Perso}{% \RedactionReciPythagore% }{% Dans le triangle $#2$, $[\NomA\NomC]$ est le plus grand c\^ot\'e.% } \PfCCalculsReciPythagore[#1]{#2}{#3}{#4}{#5}% \ifboolKV[ClesPythagore]{Egalite}{% \xintifboolexpr{\PfCRetiensReciHypo==\PfCRetiensReciSomme}{% Comme $\NomA\NomC^2=\NomA\NomB^2+\NomB\NomC^2$, alors l'\'egalit\'e de Pythagore est v\'erifi\'ee. Donc le triangle $#2$ est rectangle en $\NomB$.}{% Comme $\NomA\NomC^2\not=\NomA\NomB^2+\NomB\NomC^2$, alors l'\'egalit\'e de Pythagore n'est pas v\'erifi\'ee. Donc le triangle $#2$ n'est pas rectangle. }% }{% \xintifboolexpr{\PfCRetiensReciHypo==\PfCRetiensReciSomme}{% Comme $\NomA\NomC^2=\NomA\NomB^2+\NomB\NomC^2$, alors le triangle $#2$ est rectangle en $\NomB$ d'apr\`es la r\'eciproque du th\'eor\`eme de Pythagore.}{% Comme $\NomA\NomC^2\not=\NomA\NomB^2+\NomB\NomC^2$, alors le triangle $#2$ n'est pas rectangle\ifboolKV[ClesPythagore]{Faible}{.}{ d'apr\`es la contrapos\'ee du th\'eor\`eme de Pythagore.} }% }% }% \end{multicols} }{% \ifboolKV[ClesPythagore]{AllPerso}{% \RedactionReciPythagore% \RedactionCalculsReciPythagore% \RedactionConclusionReciPythagore% }{% \ifboolKV[ClesPythagore]{Perso}{\RedactionReciPythagore}{% Dans le triangle $#2$, $[\NomA\NomC]$ est le plus grand c\^ot\'e.% } \PfCCalculsReciPythagore[#1]{#2}{#3}{#4}{#5}% \ifboolKV[ClesPythagore]{Egalite}{% \xintifboolexpr{\PfCRetiensReciHypo==\PfCRetiensReciSomme}{% Comme $\NomA\NomC^2=\NomA\NomB^2+\NomB\NomC^2$, alors l'\'egalit\'e de Pythagore est v\'erifi\'ee. Donc le triangle $#2$ est rectangle en $\NomB$.}{% Comme $\NomA\NomC^2\not=\NomA\NomB^2+\NomB\NomC^2$, alors l'\'egalit\'e de Pythagore n'est pas v\'erifi\'ee. Donc le triangle $#2$ n'est pas rectangle. }% }{% \xintifboolexpr{\PfCRetiensReciHypo==\PfCRetiensReciSomme}{% Comme $\NomA\NomC^2=\NomA\NomB^2+\NomB\NomC^2$, alors le triangle $#2$ est rectangle en $\NomB$ d'apr\`es la r\'eciproque du th\'eor\`eme de Pythagore.}{% Comme $\NomA\NomC^2\not=\NomA\NomB^2+\NomB\NomC^2$, alors le triangle $#2$ n'est pas rectangle\ifboolKV[ClesPythagore]{Faible}{.}{ d'apr\`es la contrapos\'ee du th\'eor\`eme de Pythagore.} }% }% }% }% }% }{% % [xlop] param\`etres de calcul \opcopy{#3}{A1}% \opcopy{#4}{A2}% \opcopy{\useKV[ClesPythagore]{Precision}}{pres}% \xintifboolexpr{#3<#4 || #3==#4}{ \xdef\PetitCote{#3}% \xdef\MoyenCote{#4}% \xdef\GrandCote{}% }{% \xdef\GrandCote{#3}% \xdef\MoyenCote{#4}% \xdef\PetitCote{}% Pour la personnalisation } % On retient les noms des sommets \StrMid{#2}{1}{1}[\NomA]% \StrMid{#2}{2}{2}[\NomB]% \StrMid{#2}{3}{3}[\NomC]% \xdef\NomTriangle{\NomA\NomB\NomC}% \xdef\NomAngleDroit{\NomB}% \xdef\NomSommetA{\NomA}% \xdef\NomSommetC{\NomC}% % On trace une figure ou pas ? \ifboolKV[ClesPythagore]{FigureSeule}{% \xintifboolexpr{#3<#4 || #3==#4}{% \xdef\ResultatPytha{\fpeval{round(sqrt(#3^2+#4^2),\useKV[ClesPythagore]{Precision})}}% }{% \xdef\ResultatPytha{\fpeval{round(sqrt(#3^2-#4^2),\useKV[ClesPythagore]{Precision})}}% }% \MPFigurePytha{\NomA}{\NomB}{\NomC}{#3}{#4}{\useKV[ClesPythagore]{Angle}} }{% \ifboolKV[ClesPythagore]{Figure}{% \begin{multicols}{2}% {\em La figure est donn\'ee \`a titre indicatif.}% \[\MPFigurePytha{\NomA}{\NomB}{\NomC}{#3}{#4}{\useKV[ClesPythagore]{Angle}}\] \par\columnbreak\par% % On d\'emarre la r\'esolution \ifboolKV[ClesPythagore]{AllPerso}{% \RedactionPythagore% \RedactionCalculsPythagore% }{% \ifboolKV[ClesPythagore]{Perso}{% \RedactionCalculsPythagore% }{% \ifboolKV[ClesPythagore]{Egalite}{Comme le triangle $#2$ est rectangle en $\NomB$, alors l'\'egalit\'e de Pythagore est v\'erifi\'ee :}{Dans le triangle $#2$ rectangle en $\NomB$, le th\'eor\`eme de Pythagore permet d'\'ecrire :% }% } \CalculsPythagore[#1]{#2}{#3}{#4}{} }% \end{multicols} }{% % On d\'emarre la r\'esolution \ifboolKV[ClesPythagore]{SansMots}{% \MPFigurePythaSansMots{\NomA}{\NomB}{\NomC}{#3}{#4}{\useKV[ClesPythagore]{Angle}}% }{% \ifboolKV[ClesPythagore]{AllPerso}{% \RedactionPythagore% \RedactionCalculsPythagore% }{% \ifboolKV[ClesPythagore]{Perso}{\RedactionPythagore}{\ifboolKV[ClesPythagore]{Egalite}{Comme le triangle $#2$ est rectangle en $\NomB$, alors l'\'egalit\'e de Pythagore est v\'erifi\'ee :}{Dans le triangle $#2$ rectangle en $\NomB$, le th\'eor\`eme de Pythagore permet d'\'ecrire :% }}% \CalculsPythagore[#1]{#2}{#3}{#4}{} }% }% }% }% }% }%