\documentclass{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
vardef invert(expr P, C) =
save o; pair o; o = 1/2[point 0 of C, point 4 of C];
save r; numeric r; r = abs (point 0 of C - o);
save s; numeric s; s = abs (P - o);
o + (P - o) * r / s * r / s
enddef;
beginfig(1);
path C, L;
numeric r;
r = 89;
C = fullcircle scaled 2r;
L = (up--down) scaled 120 shifted 184 right rotated 38;
pair a, b, o;
a = point 0 of L; b = point 1 of L; o = center C;
pair P, Q, R, P', m;
P = whatever[a, b]; o - P = whatever * (a - b) rotated 90;
Q = C intersectionpoint halfcircle zscaled (P-o) shifted 1/2[P, o];
R = C intersectionpoint halfcircle zscaled (o-P) shifted 1/2[P, o];
P' = whatever[o, P] = whatever [Q, R];
draw P -- Q -- o -- R -- cycle withcolor 1/2 white;
draw Q--R; draw o -- P;
draw L withcolor 2/3 blue;
draw C dashed evenly scaled 1/2 withcolor 1/2[2/3 blue, white];
dotlabel.top("$Q$", Q);
dotlabel.lrt("$R$", R);
dotlabel.urt("$P$", P);
dotlabel.llft("$O$", o); fill fullcircle scaled 3/4 dotlabeldiam shifted o withcolor white;
label.lft("$r$", 1/2[o, Q]);
drawdot P' withpen pencircle scaled dotlabeldiam;
label("$P'$", P' shifted 10 dir 68) withcolor 2/3 blue;
drawoptions(withcolor 2/3 blue);
label.bot("\textit{circle of inversion}", point 6 of C);
label.top(TEX("\textit{polar}") rotated angle direction 0 of L, point 1/4 of L);
z0 = P' + 20 dir -20; draw z0 -- P' cutafter fullcircle scaled 8 shifted P'
withpen pencircle scaled 1/4;
label.rt("\textit{pole}", z0);
drawoptions(withpen pencircle scaled 2 withcolor 2/3 blue);
for i = 1/8, 1/4, 3/8, 5/8, 3/4, 7/8:
draw point i of L;
draw invert(point i of L, C);
endfor
drawoptions();
label("$\displaystyle {r\over OP} = {OP' \over r}$", 1/2[point 0 of C, point 1 of L] + 12 down);
label("$\displaystyle r^2 = OP \times OP'$", 1/2[point 0 of C, point 1 of L] + 36 down);
