Problem

Source: Romanian IMO TST 2006, day 3, problem 3

Tags: geometry, incenter, circumcircle, power of a point, radical axis, geometry proposed



Let γ be the incircle in the triangle A0A1A2. For all i{0,1,2} we make the following constructions (all indices are considered modulo 3): γi is the circle tangent to γ which passes through the points Ai+1 and Ai+2; Ti is the point of tangency between γi and γ; finally, the common tangent in Ti of γi and γ intersects the line Ai+1Ai+2 in the point Pi. Prove that a) the points P0, P1 and P2 are collinear; b) the lines A0T0, A1T1 and A2T2 are concurrent.