SHARYGIN 2009 Problem 17

Given triangle $ABC$ and two points $X$ , $Y$ not lying on its circumcircle. Let $A_1$ , $B_1$ , $C_1$ be the projections of $X$ to $BC$ , $CA$ , $AB$ , and $A_2$ , $B_2$ , $C_2$ be the projections of $Y$ . Prove that the perpendiculars from $A_1$ , $B_1$ , $C_1$ to $B_2C_2$ , $C_2A_2$ , $A_2B_2$ , respectively, concur if and only if line $XY$ passes through the circumcenter of $ABC$ .