Let $I, O$ be the incenter, circumcenter of triangle $ABC$ and $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 $X, Y$ and $Z$. Prove that the circumcenter of triangle $XYZ$ coincides with $O$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$