beginfig(1); %representation d'un plan numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %on place les points du parallèlogramme représentant le plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; %on trace le parallèlogramme draw z0--z2; draw z2--z3; %on donne un effet d'epaisseur en changeant l'epaisseur du stylo pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; %on rechange l'epaisseur du stylo pickup pencircle scaled 0.5pt; %on indique le nom du plan label.urt(btex $P$ etex, z0+(0.1u,0u)); endfig; beginfig(2); %intersection de deux plans numeric u; pair t,r; transform T,S; path p[],q[]; u= 1cm; %Tracé de P t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %Tracé de Q z5 = 0.5[z0,z1]; z6 = z5 transformed S; z7 = (2.5u,-1u); z8 = z7 transformed S; z5 = 0.5[z9,z7]; z10= z9 transformed S; draw z5--z6; %ici on trace la droite d'intersection draw z9--z7; draw z9--z10; draw z10--z6; draw z6--z3; draw z6--z8 dashed evenly; p1 = z2--z6; q1= z9--z10; z11 = p1 intersectionpoint q1; draw z11--z6 dashed evenly; draw z2--z11; p2 = z7--z8; q2= z5--z1; z12 = p2 intersectionpoint q2; draw z12--z8 dashed evenly; draw z7--z12; label.rt(btex $Q$ etex, z9); %nom de la droite d'intersection label.rt(btex $d$ etex, 0.5[z5,z6]); endfig; beginfig(3); %representation d'un plan defini par trois points numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %tracé du plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %tracé des points dotlabel.top(btex $A$ etex, (2u,1.5u)); dotlabel.top(btex $B$ etex, (1u,.5u)); dotlabel.top(btex $C$ etex, (3u,1u)); endfig; beginfig(4); %representation d'un plan defini par deux droites secantes numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %tracé du plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %tracé des droites z4 = (0.8u,1u); z5 =(4u,1.5u);draw z4--z5; z6 = (1u,1.6u);z7=(3.6u,0.4u);draw z6--z7; label.rt(btex $d$ etex, z5); label.rt(btex $d'$ etex, z7); endfig; beginfig(5); %representation d'un plan defini par un point et une droite numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %tracé du plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %le point et la droite dotlabel.top(btex $A$ etex, (2u,1.5u)); z4 = (1u,.5u); z5 = (3u,1u); draw z4--z5; label.top(btex $d$ etex, z5); endfig; beginfig(6); %representation d'un plan defini par deux droites parallèles numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %tracé du plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %les deux droites z4 = (1u,1.3u); z5 =(3u,1.8u);draw z4--z5; z6 = (1.5u,0.2u);z7 = z6 shifted (z5-z4);draw z6--z7; label.rt(btex $d$ etex, z5); label.rt(btex $d'$ etex, z7); endfig; beginfig(7); %representation de deux droites non parallelles et sans point commun numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; path p[]; %tracé du plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %tracé de la droite z4 = (0.5u,.5u); z5 = (4u,1u); draw z4--z5; label.top(btex $d$ etex, z5); %tracé du point z6 = (2u,1.5u); z7 = (0.2u,1u); dotlabel.rt(btex $A$ etex, z6); %on met en place les pointillés z9 = z6 shifted 2z7; z10 = z6 shifted -3z7; p1 = z6--z10; p2 = z0--z1; z11 = p1 intersectionpoint p2; draw z6--z11 dashed evenly; draw z11--z10; draw z6--z9; %le nom de la droite label.rt(btex $d'$ etex, z10); endfig; beginfig(8); %representation d'un plan numeric u; pair t,r; transform T,S; path p[],q[]; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %Trace de P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %Trace de Q z4=(0u,-2.2u); z5 = z4 transformed T; z6 = z4 transformed S; z7 = z4 transformed T transformed S; draw z4--z6; pickup pencircle scaled 2pt; draw z4--z5; draw z5--z7; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z4+(0.1u,0u)); %Plan d'intersection z8=1/5[z0,z1]+1/2[z0,z2]; z9=3/5[z4,z5]+1/2[z6,z4]; draw z8--z9; z10 = z8 transformed S; draw z8--z10; z11 = z9 transformed S; p1 = z10--z11; q1 = z2--z3; q2 = z0--z1; q3 = z6--z7; z12 = p1 intersectionpoint q1; z13 = p1 intersectionpoint q2; z14 = p1 intersectionpoint q3; draw z10--z12; draw z12--z13 dashed evenly;draw z13--z14;draw z14--z11 dashed evenly; p2 = z8--z10; z15 = p2 intersectionpoint q1; draw z2--z15; draw z15--z12 dashed evenly;draw z12--z3; p3 = z8--z9; z16 = p3 intersectionpoint q2; z17 = p3 intersectionpoint q3; q4 = z4--z5; z18 = p3 intersectionpoint q4; draw z16--z12; draw z6--z17;draw z17--z14 dashed evenly;draw z14--z7; draw z18--z14; p4 = z9--z11; z19 = p4 intersectionpoint q4; draw z9--z19;draw z19--z11 dashed evenly; %nom des droites d'intersection label.rt(btex $d$ etex, 0.5[z12,z16]); label.rt(btex $d'$ etex, 0.5[z18,z14]); endfig; beginfig(9); %droite parallele a deux plans secants numeric u; pair t,r; transform T,S; path p[]; u= 1cm; t=(4u,1u); r=(0u,2u); T = identity shifted t; S = identity shifted r; %Plan P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %Plan Q z4=0.5[z0,z1]; z5 = z4 transformed S; draw z4--z5; label.lft(btex $\Delta$ etex,0.5[z4,z5]); z6 = z4 shifted (1.5u,-1u); z4=0.5[z6,z7]; z8 = z6 transformed S; z9 = z7 transformed S; draw z0--z4;draw z4--z6;draw z6--z8;draw z8--z9;draw z4--z7 dashed evenly; p1= z4--z1; p2 = z6--z8; z10 = p1 intersectionpoint p2; draw z4--z10 dashed evenly;draw z10--z1; p3 = z2--z3; p4 = z9--z7; z11 = p3 intersectionpoint p4; draw z11--z9;draw z11--z7 dashed evenly; draw z7--z4 dashed evenly; label.lft(btex $Q$ etex, z6+(0u,0.5u)); %droite parallele aux deux plans z12 = (-1u,0u); z13 = z12 shifted (0u,3u); draw z13--z12; label.lft(btex $d$ etex, z12); endfig; beginfig(10); %theoreme du toit numeric u; pair t,r,v; transform T,S,V; path p[]; u= 1cm; t=(3u,2u); r=(-2u,1.5u);v=(-3u,0u); T = identity shifted t; S = identity shifted r; V = identity shifted v; z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed S transformed T; z4 = z0 transformed V; z5 = z0 transformed V transformed T; draw z0--z1 withcolor blue; draw z2--z3 withcolor red; z101=0.1[z0,z1]; z10=0.9[z0,z1]; z23 = z101 transformed S; z32=z10 transformed S; z45 = z101 transformed V; z54=z10 transformed V; %pour un exemple de mise en couleur draw z101--z23 withcolor green; draw z10--z32 withcolor green; draw z23--z45 withcolor green; draw z32--z54 dashed evenly withcolor green; p1 = z101--z23; p2 = z4--z5; z11 = p1 intersectionpoint p2; draw z4--z11 withcolor blue; draw z11--z5 dashed evenly withcolor blue; label.rt(btex $d$ etex, z0); label.rt(btex $d'$ etex, z4); label.top(btex $\Delta$ etex, z2); endfig; beginfig(11); % plans paralleles a l'aide de droites secantes numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %premier plan P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droites secantes z4 = (0.8u,1u); z5 =(4u,1.5u);draw z4--z5; z6 = (1u,1.6u);z7=(3.6u,0.4u);draw z6--z7; label.rt(btex $d$ etex, z5); label.rt(btex $d'$ etex, z7); %deuxieme plan Q z10=(0u,-2.5u); z11 = z10 transformed T; z12 = z10 transformed S; z13 = z10 transformed T transformed S; draw z10--z12; draw z12--z13; pickup pencircle scaled 2pt; draw z10--z11; draw z11--z13; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z10+(0.1u,0u)); %droites secantes z14 = (0.8u,-1.5u); z15 =(4u,-1u);draw z14--z15; z16 = (1u,-0.9u);z17=(3.6u,-2.1u);draw z16--z17; label.rt(btex $d_{1}$ etex, z15); label.rt(btex $d'_{1}$ etex, z17); endfig; beginfig(12); % parallelisme plan et droite numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droite dans le plan z4 = (1u,.5u); z5 = (4u,1u); draw z4--z5; label.top(btex $d'$ etex, z4); %la droite en dehors du plan z6 = (1u,2.5u); z7 = z6 shifted z5-z4; draw z6--z7; label.top(btex $d$ etex, z6); endfig; beginfig(13); %droites orthogonales numeric u; pair t,r; transform T,S; u= 1cm; t=(2u,3u); r=(0u,2u); T = identity shifted t; S = identity shifted r; %les droites perpendiculaires en I z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; draw z0--z1; draw z0--z2; label.bot(btex $I$ etex, z0); %marquage de l'angle droit z3=0.1[z0,z1];z4=0.1[z0,z2]; z5=z3 shifted z4-z0; draw z3--z5; draw z4--z5; %droites d et delta z6=(0u,-1u); z7 = z6 transformed T; draw z7--z6; label.bot(btex $\Delta$ etex, z7); z8=(-1u,-1u); z9= z8 transformed S; label.lft(btex $d$ etex, z9); draw z9--z8; endfig; beginfig(14); %droite orthogonale a un plan numeric u; pair t,r; transform T,S; path p[]; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %plan P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droites secantes z4 = (0.8u,1u); z5 =(4u,1.5u);draw z4--z5; z6 = (1u,1.6u);z7=(3.6u,0.4u);draw z6--z7; label.rt(btex $d$ etex, z5); label.rt(btex $d'$ etex, z7); p1 = z4--z5;p2 = z6--z7; %droite orthogonale z8 = p1 intersectionpoint p2; z9 = z8 shifted (0u,2u); label.rt(btex $\Delta$ etex, z9); draw z8--z9; z10 = z8 shifted (0u,-2u); p3 = z9--z10;p4 = z0--z1; z11 = p3 intersectionpoint p4; draw z8--z11 dashed evenly;draw z11--z10; %marquage des angles droits z12=0.1[z8,z7];z13=0.1[z8,z9]; z14= z13 shifted z12-z8; draw z12--z14; draw z13--z14; z15=0.2[z8,z4];z16=0.1[z8,z9]; z17= z16 shifted z15-z8; draw z15--z17; draw z16--z17; endfig; beginfig(15); %plans paralleles et droite orthogonale numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; path p[]; %premier plan et droites z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droites secantes z4 = (0.8u,1u); z5 =(4u,1.5u);draw z4--z5; z6 = (1u,1.6u);z7=(3.6u,0.4u);draw z6--z7; label.rt(btex $d$ etex, z5); label.rt(btex $d'$ etex, z7); %deuxieme plan et droites z10=(0u,-2.5u); z11 = z10 transformed T; z12 = z10 transformed S; z13 = z10 transformed T transformed S; draw z10--z12; draw z12--z13; pickup pencircle scaled 2pt; draw z10--z11; draw z11--z13; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z10+(0.1u,0u)); %droites secantes z14 = (0.8u,-1.5u); z15 =(4u,-1u);draw z14--z15; z16 = (1u,-0.9u);z17=(3.6u,-2.1u);draw z16--z17; label.rt(btex $d_{1}$ etex, z15); label.rt(btex $d'_{1}$ etex, z17); %droite orthogonale p1=z4--z5;p2=z6--z7; p3=z14--z15;p4=z16--z17; p5=z0--z1;p6=z10--z11; z20 = p1 intersectionpoint p2; z21 = p3 intersectionpoint p4; p7 = z20--z21; z22 = p7 intersectionpoint p5; z24 = z20 shifted (0u,1.5u); z25 = z21 shifted (0u,-1.5u); p8 = z21--z25; z23 = p8 intersectionpoint p6; draw z24--z20;draw z20--z22 dashed evenly; draw z22--z21; draw z21--z23 dashed evenly; draw z23--z25; label.rt(btex $\Delta$ etex, z24); %marquage des angles droits z30=0.2[z20,z4];z31=0.1[z20,z24]; z32 = z31 shifted z30-z20; draw z32--z30; draw z32--z31; z40=0.2[z21,z14];z41=0.1[z21,z22]; z42 = z41 shifted z40-z21; draw z42--z40; draw z42--z41; z50=0.1[z20,z7];z51=0.1[z20,z24]; z52 = z51 shifted z50-z20; draw z52--z50; draw z52--z51; z60=0.1[z21,z17];z61=0.1[z21,z22]; z62 = z61 shifted z60-z21; draw z62--z60; draw z62--z61; endfig; beginfig(16); %droites orthogonales a deux plans paralleles numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; path p[]; %premier plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %deuxieme plan z10=(0u,-2.5u); z11 = z10 transformed T; z12 = z10 transformed S; z13 = z10 transformed T transformed S; draw z10--z12; draw z12--z13; pickup pencircle scaled 2pt; draw z10--z11; draw z11--z13; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z10+(0.1u,0u)); %premiere droite z20 = (2u,1u);z26 = z20 shifted (0u,-2.5u); z21 = z20 shifted (0u,2u); z22 = z20 shifted (0u,-4.5u); p1 = z21--z22; p2 = z0--z1; p3 = z10--z11; z24 = p1 intersectionpoint p2; z25 = p1 intersectionpoint p3; draw z21--z20;draw z20--z24 dashed evenly;draw z24--z26;draw z26--z25 dashed evenly; draw z25--z22; %deuxieme droite z30 = (3u,1.5u);z36 = z30 shifted (0u,-2.5u); z31 = z30 shifted (0u,2u); z32 = z30 shifted (0u,-4.5u); p11 = z31--z32; p12 = z0--z1; p13 = z10--z11; z34 = p11 intersectionpoint p12; z35 = p11 intersectionpoint p13; draw z31--z30;draw z30--z34 dashed evenly;draw z34--z36;draw z36--z35 dashed evenly; draw z35--z32; %marquage des angles droits z40=0.1[z20,z30];z41=0.1[z20,z21]; z42 = z41 shifted z40-z20; draw z42--z40; draw z42--z41; z50=0.1[z30,z20];z51=0.1[z30,z31]; z52 = z51 shifted z50-z30; draw z52--z50; draw z52--z51; z60=0.1[z26,z36];z61=0.1[z26,z24]; z62 = z61 shifted z60-z26; draw z62--z60; draw z62--z61; z70=0.1[z36,z26];z71=0.2[z36,z34]; z72 = z71 shifted z70-z36; draw z72--z70; draw z72--z71; draw z20--z30; draw z26--z36; endfig; beginfig(17); %intersection de deux plans perpendiculaires numeric u; pair t,r; transform T,S; path p[],q[]; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %Tracé de P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %Tracé de Q z5 = 0.5[z0,z1]; z6 = z5 transformed S; z7 = z5 shifted (0u,-2.5u); z8 = z7 transformed S; z5 = 0.5[z9,z7]; z10= z9 transformed S; draw z5--z6; draw z9--z7; draw z9--z10; draw z10--z6; draw z6--z3; draw z7--z8; p1 = z2--z6; q1= z5--z9; z11 = p1 intersectionpoint q1; draw z11--z6 dashed evenly; draw z2--z11; p2 = z6--z8; q2= z5--z1; z12 = p2 intersectionpoint q2; draw z12--z6 dashed evenly; draw z8--z12; label.rt(btex $Q$ etex, z9); %nom de la droite d'intersection label.rt(btex $d$ etex, 0.6[z5,z6]); %droites orthogonales z13=0.5[z5,z6]; z14 = z13 shifted (0u,2u); z15 = z13 shifted (0u,-2u); p3 = z0--z1; q3 = z13--z15; z16 = p3 intersectionpoint q3; draw z14--z13; draw z13--z16 dashed evenly; draw z16--z15; %marquage de l'angle droit z20=0.2[z13,z6];z21=0.2[z13,z14]; z22 = z21 shifted z20-z13; draw z22--z20; draw z22--z21; z17 = z13 shifted 0.8(z1-z5); z18 = z13 shifted 0.8(z0-z5); p4 = z13--z18; z19 = q1 intersectionpoint p4; draw z18--z19; draw z19--z13 dashed evenly; draw z13--z17; %marquage de l'angle droit z30=0.1[z13,z18];z31=0.1[z13,z14]; z32 = z31 shifted z30-z13; draw z32--z30; draw z32--z31; endfig; beginfig(18); %plans perpendiculaires a un meme plan numeric u; pair t,r; transform T,S; path p[],q[]; u= 1cm; t=(5u,0u); r=(1u,2.3u); T = identity shifted t; S = identity shifted r; %plan Q z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z0+(0.1u,0u)); %les plans perpendiculaires z4= 1/4[z0,z1]; z5= 3/4[z0,z1]; z6= 1/2[z4,z5]; z7 = z6 shifted (0u,1u); draw z4--z7; draw z5--z7; z8 = z4 shifted (0u,2u); z9 = z5 shifted (0u,2u); z10 = z7 shifted (0u,2u); z4=0.5[z8,z11];z5=0.5[z12,z9];z7=0.5[z13,z10]; draw z8--z11;draw z8--z10; draw z12--z9;draw z9--z10; draw z7--z10; draw z7--z6 dashed evenly; draw z6--z13; draw z11--z13; draw z13--z12; p1 = z2--z3; q1 = z8--z10;q2=z9--z10; z14= p1 intersectionpoint q1; z15= p1 intersectionpoint q2; draw z2--z14;draw z14--z15 dashed evenly; draw z15--z3; %marquage des angles droits; z20=0.1[z7,z10];z21=0.2[z7,z4]; z22 = z21 shifted z20-z7; draw z22--z20; draw z22--z21; z30=0.1[z7,z10];z31=0.2[z7,z5]; z32 = z31 shifted z30-z7; draw z32--z30; draw z32--z31; label.bot(btex $P$ etex, z11+(0.2u,0.6u)); label.bot(btex $P'$ etex, z12+(-0.2u,0.8u)); endfig; end