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\documentclass[border=1mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
vardef radical_axis(expr ca, cb) =
numeric t, d, ra, rb;
ra = abs(center ca - point 0 of ca);
rb = abs(center cb - point 0 of cb);
d = abs(center cb - center ca);
2t = 1 + (ra+rb) / d * (ra-rb) / d;
(up -- down) scaled 89
rotated angle (center cb - center ca)
shifted t[center ca, center cb]
enddef;
input tangent-times
vardef invert_point(expr P, o, r) =
save p, d; pair p; numeric d;
p = P - o; d = abs p;
if d > 0:
o + unitvector p scaled (r/d*r)
else:
errmessage("Inversion undefined at center.")
fi
enddef;
vardef pole(expr Line, Circle) =
save p, o, r; pair o, p; numeric r;
o = center Circle;
r = 1/2 abs (point 4 of Circle - point 0 of Circle);
p = whatever [point 1 of Line, point 0 of Line];
p - o = whatever * direction 0 of Line rotated 90;
invert_point(p, o, r)
enddef;
vardef polex(expr a, b, o, r) =
save p; pair p;
p = whatever [a, b];
p - o = whatever * (a-b) rotated 90;
invert_point(p, o, r)
enddef;
vardef three_point_circle(expr a,b,c) =
save m; pair m;
m = whatever [a,b] rotatedaround(.5[a,b],90)
= whatever [b,c] rotatedaround(.5[b,c],90);
fullcircle scaled 2 length(m-a) shifted m
enddef;
vardef through(expr a, b, o) =
save d; numeric d; d = abs(a-b);
(1+o/d)[b, a] -- (1+o/d)[a, b]
enddef;
beginfig(1);
path c[]; numeric r[];
z1 = origin; r1 = 101;
z2 = 233 right rotated 4; r2 = 53;
z3 = 209 right rotated -42; r3 = 31;
forsuffixes $=1, 2, 3:
c$ = fullcircle scaled 2 r$ shifted z$;
endfor
pair ecs[], ics[];
for i=1 upto 3:
numeric j, k;
j = i mod 3 + 1;
k = 10i + j;
ics[k] = (r[i]/(r[i]+r[j]))[z[i], z[j]];
ecs[k] = (r[i]/(r[i]-r[j]))[z[i], z[j]];
endfor
path a[];
a1 = radical_axis(c1, c2);
a2 = radical_axis(c2, c3);
a3 = radical_axis(c3, c1);
z0 = whatever [point 0 of a1, point 1 of a1]
= whatever [point 0 of a2, point 1 of a2];
z11 = polex(ecs31, ecs12, z1, r1);
z21 = polex(ecs31, ecs12, z2, r2);
z31 = polex(ecs31, ecs12, z3, r3);
z12 = c1 intersectionpoint (z0 -- z11);
z22 = c2 intersectionpoint (z0 -- z21);
z32 = c3 intersectionpoint (z0 -- z31);
z13 = c1 intersectionpoint (z11 -- 8[z0,z11]);
z23 = c2 intersectionpoint (z21 -- 8[z0,z21]);
z33 = c3 intersectionpoint (z31 -- 8[z0,z31]);
z14 = whatever[ecs12, ecs31] = whatever[z1, z11];
z24 = whatever[ecs12, ecs31] = whatever[z2, z21];
z34 = whatever[ecs12, ecs31] = whatever[z3, z31];
drawoptions(withcolor 3/4[blue, white]);
draw c1; draw c2; draw c3;
drawoptions(withcolor 1/4[blue, white]);
label.urt(btex $C_1$ etex, point 1 of c1);
label.top(btex $C_2$ etex, point 2 of c2);
label.rt (btex $C_3$ etex, point 1/2 of c3);
draw ecs12 -- ecs31 -- ecs23;
drawoptions(withpen pencircle scaled 1/4 withcolor 3/4[2/3 blue, white]);
draw a1; draw a2; draw a3;
draw through(z1, z14, 6);
draw through(z2, z24, 6);
draw through(z3, z34, 6);
drawoptions(withpen pencircle scaled 1/4 withcolor 1/2 white);
draw through(z0, z13, 6);
draw through(z0, z23, 6);
draw through(z0, z33, 6);
drawoptions(withcolor 1/256(203, 92, 13));
drawdot z0 withpen pencircle scaled 3/2 dotlabeldiam;
z99 = z0 shifted 24 dir -6;
label.rt("\vbox{\openup-4pt\halign{\hss #\hss\cr Radical\cr centre\cr}}", z99);
interim ahangle := 20; drawarrow z99 -- z0
cutafter fullcircle scaled 16 shifted z0
withpen pencircle scaled 1/3;
drawoptions(withcolor 1/256(239, 114, 21));
dotlabel.urt("\small Pole", z11);
dotlabel.ulft("\small Pole", z21);
dotlabel.urt("\small Pole", z31);
drawoptions(withcolor 2/3 red);
draw three_point_circle(z12, z22, z32);
draw three_point_circle(z13, z23, z33);
drawdot z12 withpen pencircle scaled 3/4 dotlabeldiam;
drawdot z22 withpen pencircle scaled 3/4 dotlabeldiam;
drawdot z32 withpen pencircle scaled 3/4 dotlabeldiam;
drawdot z13 withpen pencircle scaled 3/4 dotlabeldiam;
drawdot z23 withpen pencircle scaled 3/4 dotlabeldiam;
drawdot z33 withpen pencircle scaled 3/4 dotlabeldiam;
drawoptions(withcolor 1/2[blue, white]);
drawdot z1 withpen pencircle scaled dotlabeldiam;
drawdot z2 withpen pencircle scaled dotlabeldiam;
drawdot z3 withpen pencircle scaled dotlabeldiam;
drawdot z14 withpen pencircle scaled dotlabeldiam;
drawdot z24 withpen pencircle scaled dotlabeldiam;
drawdot z34 withpen pencircle scaled dotlabeldiam;
draw thelabel.top(btex Axis of similitude, as polar etex, origin)
rotated angle (ecs12 - ecs31)
shifted 3/4[ecs31, ecs12];
drawoptions();
endfig;
\end{mplibcode}
\end{document}
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