- Apr 22, 2014AntreasThey are orthologic for any point P (not only for I)The triangles ABC, bounded by (R1,R2,R3) are orthologic.R2 = the radical axis of the circumcircles of ABC and BC'A'R1 = the radical axis of the circumcircles of ABC and AB'C'Denote:Let ABC be a triangle and A'B'C' the pedal triangle of I.

Let be ABC a scalene triangle with circumcircle Gamma and let D,E,F be the points where its incircle meets BC, AC, AB respectively. Let the circumcircles of AEF, BFD and CDE meet Gamma a second time at X,Y,Z respectively. Prove that the perpendiculars from A.B,C to AX, BY, CZ respectively are concurrent.

Proposed by Michael Kural.Translation into triangle geometry language :-)

R3 = the radical axis of the circumcircles of ABC and CA'B'