KURSCHAK 1977 Problem 2

$ABC$ is a triangle with orthocenter $H$ . The median from $A$ meets the circumcircle again at $A_1$ , and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$ . The points $B_2$ and $C_2$ are defined similarly. Show that $H$ , $A_2$ , $B_2$ and $C_2$ lie on a circle.