u:=2cm; beginfig(1); numeric theta; theta:=20; z0=origin; z2-z0=2u*left; pair delta; delta=unitvector (z2-z0 rotated theta) * 3; z1-z0=((z2-z0) rotated -theta) + delta; z3-z0=(z2-z0) rotated theta + 2delta; draw z0--z1--z2--z0--z3--z2 withpen pencircle scaled .7pt; label.rt("0",z0); label.urt("1",z1); label.lft("n-2",z2); label.llft("n-1",z3); currentpicture:=currentpicture shifted -z3; endfig; beginfig(2); numeric theta; theta:=45; z0=origin; z2-z0=2u*left; pair delta; delta=unitvector (z2-z0 rotated theta); z1-z0=((z2-z0) rotated -theta) + 50delta; z3-z0=((z1-z0) rotated 10) -5delta; z4-z0=((z3-z0) rotated 10) -5delta; z2'=whatever[z3,z4]; z2'=whatever[z0,z2]; z5=whatever[z3,z4]; z5=whatever[z2,z0]; pickup pencircle scaled .6pt; draw z0--z1; draw z3--z4; draw z1--z3 dashed dashpattern (on 1bp off 1bp); draw z0--z2; drawarrow z0--z4; label.top(btex $r$ etex, .5[z4,z0]); draw z1--z2 dashed evenly; draw z3--z2 dashed evenly; draw z4--z2 dashed evenly; path circ; circ=halfcircle scaled 2length(z4-z0) rotated 90; draw circ cutbefore (z0--z1) cutafter (z0--(z2'-(20pt,0))); draw z2'--z2 dashed evenly; draw z5--z4 dashed evenly; z6-z0=(z2'-z0) rotated 10; z7=(z5+z2+z4)/3; z8=whatever[z0,z6]; z8=whatever[z7,z2]; z9=whatever[z2',z4]; z9=whatever[z7,z2]; draw z8--z9; draw z0--z8; fill z9--z2--z4--cycle withcolor .7white; label.bot(btex $n-1$ etex,z8); label.rt("0",z0); label.urt("1",z1); label.llft(btex $n-2$ etex,z2); dotlabel.lft(btex $k-1$ etex,z3); dotlabel.lft(btex $k$ etex,z4); drawdot (z9+z2+z4)/3 withpen pencircle scaled 2pt; endfig; beginfig(3); numeric theta; theta:=10; z0=origin; z2-z0=2u*left; pair delta; delta=unitvector ((z2-z0) rotated theta) * 3; z3-z0=((z2-z0) rotated -theta); z1-z0=(z2-z0) rotated theta + 2delta; for i:=3 upto 6: z[i]-z0=(z[i-1]-z0) rotated -theta; draw z0--z[i]; endfor; draw z0--z2; path arc; arc = halfcircle scaled 2length(z2-z0) rotated 90 cutbefore (z0--(2[z0,z6])) cutafter (z0--z2); draw arc; label.rt("0", z0); label.urt("x", z6); label.lft(btex $1'$ etex, z5); label.lft(btex $2'$ etex, z4); label.lft(btex $n-2$ etex, z3); label.lft(btex $n-1$ etex, z2); label.top(btex \vdots etex, .8[z0,z3]); endfig; beginfig(4); numeric theta; theta:=20; z0=origin; z2-z0=2u*left; for i:=3 upto 5: z[i]-z0=(z[i-1]-z0) rotated -theta; draw z0--z[i]; endfor; draw z0--z2; draw z2--z4; draw z3--z5; path arc; arc = halfcircle scaled 2length(z2-z0) rotated 90 cutbefore (z0--z5) cutafter (z0--z2); draw arc; z6=(z0--z3) intersectionpoint (z2--z4); z7=(z0--z4) intersectionpoint (z3--z5); z8=.3[z3,z6]; z9=.7[z4,z7]; draw z2--z8--z9 dashed evenly withpen pencircle scaled .8pt; draw z3--z6 withpen pencircle scaled 2pt; label.urt(btex $r$ etex, .5[z0,z4]); %dotlabels.bot(0,2,3,4,5,6,7); endfig; end