BG-MO 2004 Problem 1

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