- Jun 28, 2014Antreasto NPC and three excircles etcIf true, then we can apply it to Circumcircle and three Lucas Circles,The lines AMa, BMb, CMc are concurrent ??Mc = the midpoint of AcBcAnother Inversion Theorem (?)Mb = the midpoint of CbAbLet (A), (B), (C) be three circles tangent all externally or internally

a circle Q at A',B',C', resp.

Denote:Ab = the inversion of A' in (B)Ac = the inversion of A' in (C)Bc = the inversion of B' in (C)Ba = the inversion of B' in (A)Ca = the inversion of C' in (A)Cb = the inversion of C' in (B)Ma = the midpoint of BaCa

[APH]Let (A), (B), (C) be three externally tangent circles.The [Apollonian] circle tangent to (A),(B),(C) internallytouches them at A',B',C', resp.Denote:Ab = the inversion of A' in (B)Ac = the inversion of A' in (C)Bc = the inversion of B' in (C)Ba = the inversion of B' in (A)Ca = the inversion of C' in (A)Cb = the inversion of C' in (B)Are the six points Ab,Ac.Bc,Ba,Bc,Ca,Cb always concyclic?In my figure they are, and the center of the circle lieson the line joining the circumcenters of ABC and A'B'C'.Is it true? And if yes, is it already known? (references?)Antreas