Problem

Source: Ukraine TST 2014 p4

Tags: geometry, circumcircle, excircle, Circumcenter



The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.