Question 110
Let $ABC$ be a scalene triangle, $I$ be its incenter and $O$ be its circumcenter. The line $IO$ intersects the lines $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Let $A_1$ be the intersection of $BE$ and $CF$. The points $B_1$ and $C_1$ are defined similarly. The incircle of $ABC$ is tangent to sides $BC$, $CA$, $AB$ at points $X$, $Y$, $Z$, respectively. Let the lines $XA_1$, $YB_1$ and $ZC_1$ intersect $IO$ at points $A_2$, $B_2$, $C_2$ respectively. Prove that the circles with diameters $AA_2$, $BB_2$ and $CC_2$ have a common point.