Can I just interject that such computations involving circles (and spheres in higher dimension) are all special cases of the "Generalised Apollonian Problem", which has a complete formal solution in terms of the Clifford algebra based on pentacyclic coordinates for plane contact (Lie-sphere) geometry. Looking ahead, can anyone come up with an algebraic proof of Euler-Chapple: (R - 2r) R = d^2 where R, r, d denote circumradius, inradius (or exradius), displacement of centres resp. of a plane Euclidean triangle? My sledgehammer attempts have so far met with humiliating defeat ... WFL On 1/28/16, Bill Gosper <billgosper@gmail.com> wrote:
I hope Henry or someone is collecting our various complex number geometricks e.g., PolygonArea[L_List] := Total[MapThread[Im[#1\[Conjugate]*#2] &, {L, RotateLeft[L]}]]/2
line segment intersect Interseg[z1_, z2_, z3_, z4_] := Block[{z12 = z1 - z2, z34 = z3 - z4}, (z34*Im[z2*z1\[Conjugate]] + z12*Im[z3*z4\[Conjugate]])/Im[z34*z12\[Conjugate]]]
incenter, circumcenter, inradius, ... .
For the Clifford circle theorems I needed second intersection: Given centers of circles though the origin z1, z2,
secondintersection[z1_, z2_] := (Conjugate[z1] z2 - Conjugate[z2] z1)/Conjugate[z1 - z2]
Is this obvious to anybody? I had to derive it with 8th grade analytic geometry. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun