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another incenter problem

Source: Bulgarian Math Olympiad MO 2004, problem 1

May 17, 2004
geometryincentercircumcirclegeometry proposed

Problem Statement

Let I I be the incenter of triangle ABC ABC, and let A1 A_1, B1 B_1, C1 C_1 be arbitrary points on the segments (AI) (AI), (BI) (BI), (CI) (CI), respectively. The perpendicular bisectors of AA1 AA_1, BB1 BB_1, CC1 CC_1 intersect each other at A2 A_2, B2 B_2, and C2 C_2. Prove that the circumcenter of the triangle A2B2C2 A_2B_2C_2 coincides with the circumcenter of the triangle ABC ABC if and only if I I is the orthocenter of triangle A1B1C1 A_1B_1C_1.
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