OMCPLP 2016 Problem 2

The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$ , a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$ , let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$ . Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.