RU-SPB 2022 Problem 115

Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$ . On the tangent drawn from point $C$ to the circle $(AB_1C_1)$ , the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$ ). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$ .