Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.
Problem
Source: MOP 2005 Homework - Black Group #16
Tags: geometry, incenter, circumcircle, search, geometry unsolved