Question 57
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 $A_{1}B_{1}C_{1}$.