gosper.org/cliff.png Anticipated by https://en.wikipedia.org/wiki/Clifford%27s_circle_theorems except for the (terse) drawing commands, for which you need circumboth[z1_, z2_, z3_] := {((z1 - z2)*Abs[z3]^2 + Abs[z2]^2*(z3 - z1) + Abs[z1]^2*(z2 - z3))/I, Abs[(z2 - z1)*(z2 - z3)*(z3 - z1)]}/(2*Im[Conjugate[z2 - z1]*(z3 - z1)]) second[z1_, z2_] := (Conjugate[z1] z2 - Conjugate[z2] z1)/Conjugate[z1 - z2] for {circumcenter, circumradius} and second intersection. ---------- Forwarded message ---------- From: Bill Gosper <billgosper@gmail.com> Date: Sat, Jan 30, 2016 at 1:43 AM Subject: Pic: Clifford's 1st thm To: mathfuneavesdroppers@googlegroups.com, rcs@xmission.com Coxeter says: Put four arbitrary black circles through a point. Set aside each in turn. For each of the four remaining triads, put a red circle through their three second intersections. The four red circles meet at a point, and red and black can be interchanged. --rwg --------------- Related stuff: clifford circle chains <https://www.google.com/search?q=clifford+circle+chains&client=safari&rls=en&biw=1873&bih=1055&tbm=isch&tbo=u&source=univ&sa=X&sqi=2&ved=0ahUKEwjMgbLgydTKAhUK5WMKHa_lB_EQsAQIKg#imgrc=r6fHrozYkNx1DM%3A> images . Clifford's 2nd thm might require animation to be visually intelligible. A difficulty is curvatures changing sign (radii blowing up).