KZ-MO 2002 Problem 5

On the plane is given the acute triangle $ABC$ . Let $A_1$ and $B_1$ be the feet of the altitudes of $A$ and $B$ drawn from those vertices, respectively. Tangents at points $A_1$ and $B_1$ drawn to the circumscribed circle of the triangle $CA_1B_1$ intersect at $M$ . Prove that the circles circumscribed around the triangles $AMB_1$ , $BMA_1$ and $CA_1B_1$ have a common point.