Middle European Mathematical Olympiad 2011 Problem T-6

Let $ABC$ be an acute triangle. Denote by $B_{0}$ and $C_{0}$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_{0}$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_{0}$. Show that the line $AX$ is perpendicular to $BC$.