SHARYGIN 2017 Problem 12

Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$ ; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$ , distinct from $B_0$ . Prove that $BQ$ is the symmedian of $ABC$ . Proposed by D.Shvetsov