\input slideshow; author("Christophe Poulain & Régis Leclercq"); title("Le Plaisir de la géométrie : fiche 310"); %%[navigation] \input navigation; couleurboutons:=(1,1,0.3); couleurfond:=(0,0,0.7); %% Choix de LaTeX verbatimtex %&latex \documentclass[a4paper]{article} \usepackage[latin1]{inputenc} \usepackage[frenchb]{babel} \usepackage[dvips]{color,graphicx} \begin{document} \footnotesize etex %%[geometrie] %input geometrie1 def orthocentre(expr p,q,r )=%(sommet-côté opposé) begingroup save $; pair $; ($-p) rotated 90=whatever*(r-q); ($-q) rotated 90=whatever*(p-r); $ endgroup enddef; def cercle(expr p,q)=%centre-rayon begingroup save $; path $; $=fullcircle scaled (2*q) shifted p; $ endgroup enddef; def droite(expr a,b)=%Points begingroup save $; path $; $=(-3)[a,b]--3[a,b]; $ endgroup enddef; def codeang(expr p,q,r,n)=%point-sommet-point(sens direct)-rayon du codage begingroup save $,cc,b,c,f,seg,sege,d,e; picture $; $=currentpicture; path cc;%cercle pour le codage cc=fullcircle scaled (2*n*unit) shifted q; pair b,c,f; path seg,sege; seg=p--q; sege=q--r; b=cc intersectionpoint seg; c=cc intersectionpoint sege; numeric d,e; d=(angle(b-q))*(length cc)/360; e=(angle(c-q))*(length cc)/360; draw subpath(d,e) of cc; $ endgroup enddef; vardef codeperp(expr a,b,c) = (b+5*unitvector(a-b))--(b+5*unitvector(a-b)+5*unitvector(c-b))--(b+5*unitvector(c-b)) enddef; def perp(expr p,q,r)=%point-droite begingroup save $,cc,ce,cd,cf; picture $; pair cc,ce; path cd,cf; $=currentpicture; cc=(r-q) rotated 90 shifted q; ce=cc shifted (p-q); cd=droite(cc shifted (p-q),q shifted (p-q)); draw cd dashed evenly withcolor blue; cf=droite(q,r); draw codeperp(ce,cf intersectionpoint cd,r); $ endgroup enddef; %%[Image] picture cp; cp=thelabel(btex \sf C.POULAIN \& R.LECLERQ -- 2001 etex scaled 0.5,(0.85lawidth,0.03laheight)); footer(image(draw cp withcolor blue;)); %% -- navigation PDF & logos def navPDFlogos = init_navigation; navigation; internavigation; enddef; extra_endfig := extra_endfig & "navPDFlogos;"; % gs will need this prologues:=2; %fond d'écran noslides := 11; def doback = background := image(drawgradient((charcode/noslides)[white,white], (charcode/noslides)[white,white]);); enddef; doback; %cadre path cadre; cadre=(0.05lawidth,0.9laheight)--(0.95lawidth,0.9laheight)--(0.95lawidth,0.98laheight)--(0.05lawidth,0.98laheight)--cycle; draw cadre; %% -- instructions color C[]; C6=0.8white; vardef instruction(expr s) = fill cadre withcolor C6; label.rt(s,p100); enddef; %Couleurs color aubergine,beige; aubergine = 3(37/256,2/256,29/256); beige = (0.77734375,0.67578125,0.4921875); C0 = (.5,1,1); C1 = (.9,.4,.5); C2 = 0.2[C0,C1]; C3 = 0.2[C1,C0]; C4 = 0.3[red,green]; C5=(.5,1,.25); %[Points] pair p[],h[],k[]; unit=1cm; p0=(0.5lawidth,0.5laheight);%O p1=(0.6lawidth,0.5laheight);%A for i:=2 upto 5 : p[i]=(p[i-1]-p0) rotated 72 shifted p0; endfor h1=orthocentre(p1,p2,p3); h2=orthocentre(p2,p3,p4); h3=orthocentre(p3,p4,p5); h4=orthocentre(p4,p5,p1); h5=orthocentre(p5,p1,p2); k1=droite(p1,p2) intersectionpoint droite(p3,p4); k2=droite(p2,p3) intersectionpoint droite(p4,p5); k3=droite(p3,p4) intersectionpoint droite(p5,p1); k4=droite(p4,p5) intersectionpoint droite(p1,p2); k5=droite(p5,p1) intersectionpoint droite(p2,p3); p100=(0.07lawidth,0.94laheight); %Animation nextfig; label(btex \Large\bf Le Plaisir de la Géométrie etex scaled 2,(.502lawidth,.548laheight)) withcolor 0.3white; blabel(btex \Large\bf Le Plaisir de la Géométrie etex scaled 2,(.50lawidth,.55laheight)); label(btex \Large\bf Fiche 310 etex scaled 2,(.502lawidth,.408laheight)) withcolor 0.3white; blabel(btex \Large\bf Fiche 310 etex scaled 2,(.5lawidth,.41laheight)); hyperdest("start"); endfig; discontinue; nextfig; instruction(btex 1. Trace un cercle de centre $O$ et de rayon $OA=4,5\,cm$. etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); endfig; continue; nextfig; instruction(btex 2. Place sur le cercle, les points $B$, $C$, $D$ et $E$ tels que $\widehat{AOB}=\widehat{BOC}=\widehat{COD}=\widehat{DOE}=72$° etex); draw p1--p0--p2; draw codeang(p1,p0,p2,0.9); labeloffset:=8pt; label.urt(btex 72° etex,p0); labeloffset:=3bp; dotlabel.top(btex $B$ etex,p2); endfig; nextfig; draw p0--p3; dotlabel.top(btex $C$ etex,p3); endfig; nextfig; draw p0--p4; dotlabel.bot(btex $D$ etex,p4); endfig; nextfig; draw p0--p5; dotlabel.bot(btex $E$ etex,p5); endfig; discontinue; nextfig; instruction(btex 3. Construire les hauteurs du triangle $ABC$ : elles se coupent en $H_1$ etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); dotlabel.top(btex $B$ etex,p2); dotlabel.top(btex $C$ etex,p3); dotlabel.bot(btex $D$ etex,p4); dotlabel.bot(btex $E$ etex,p5); draw droite(p1,p2); draw droite(p1,p3); draw droite(p3,p2); draw perp(h1,p1,p2); endfig; nextfig; draw perp(h1,p2,p3); endfig; nextfig; draw perp(h1,p3,p1); dotlabel.top(btex $H_1$ etex,h1); endfig; discontinue; nextfig; instruction(btex 4. Construis les hauteurs du triangle $BCD$ : elles se coupent en $H_2$ etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); dotlabel.top(btex $B$ etex,p2); dotlabel.top(btex $C$ etex,p3); dotlabel.bot(btex $D$ etex,p4); dotlabel.bot(btex $E$ etex,p5); dotlabel.top(btex $H_1$ etex,h1); draw droite(p3,p2); draw droite(p4,p2); draw droite(p4,p3); draw perp(h2,p2,p3); draw perp(h2,p3,p4); draw perp(h2,p4,p2); dotlabel.top(btex $H_2$ etex,h2); endfig; nextfig; instruction(btex 4.1. Les droites $(AB)$ et $(CD)$ se coupent en $K_1$. etex); draw droite(p1,p2) withcolor red; dotlabel.top(btex $K_1$ etex,k1); endfig; discontinue; nextfig; instruction(btex 5. Construis les hauteurs du triangle $CDE$ : elles se coupent en $H_3$ etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); dotlabel.top(btex $B$ etex,p2); dotlabel.top(btex $C$ etex,p3); dotlabel.top(btex $D$ etex,p4); dotlabel.bot(btex $E$ etex,p5); dotlabel.bot(btex $H_1$ etex,h1); dotlabel.top(btex $H_2$ etex,h2); dotlabel.top(btex $K_1$ etex,k1); draw droite(p3,p4); draw droite(p4,p5); draw droite(p5,p3); draw perp(h3,p3,p4); draw perp(h3,p4,p5); draw perp(h3,p5,p3); dotlabel.bot(btex $H_3$ etex,h3); endfig; nextfig; instruction(btex 5.1. Les droites $(BC)$ et $(DE)$ se coupent en $K_2$. etex); draw droite(p3,p2) withcolor red; dotlabel.top(btex $K_2$ etex,k2); endfig; discontinue; nextfig; instruction(btex 6. Construis les hauteurs du triangle $DEA$ : elles se coupent en $H_4$ etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); dotlabel.top(btex $B$ etex,p2); dotlabel.top(btex $C$ etex,p3); dotlabel.bot(btex $D$ etex,p4); dotlabel.bot(btex $E$ etex,p5); dotlabel.top(btex $H_1$ etex,h1); dotlabel.top(btex $H_2$ etex,h2); dotlabel.bot(btex $H_3$ etex,h3); dotlabel.top(btex $K_1$ etex,k1); dotlabel.top(btex $K_2$ etex,k2); draw droite(p5,p4); draw droite(p1,p5); draw droite(p4,p1); draw perp(h4,p4,p5); draw perp(h4,p5,p1); draw perp(h4,p1,p4); dotlabel.bot(btex $H_4$ etex,h4); endfig; nextfig; instruction(btex 6.1. Les droites $(CD)$ et $(EA)$ se coupent en $K_3$. etex); draw droite(p3,p4) withcolor red; dotlabel.lrt(btex $K_3$ etex,k3); endfig; discontinue; nextfig; instruction(btex 7. Construis les hauteurs du triangle $EAB$ : elles se coupent en $H_5$ etex); draw cercle(p0,abs(p0-p1)) dashed evenly; dotlabel.ulft(btex $O$ etex,p0); dotlabel.rt(btex $A$ etex,p1); dotlabel.top(btex $B$ etex,p2); dotlabel.top(btex $C$ etex,p3); dotlabel.bot(btex $D$ etex,p4); dotlabel.bot(btex $E$ etex,p5); dotlabel.top(btex $H_1