RU-SPB 2014 Problem 105

Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$ . Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$ . Tangents to $\omega$ at $E,F$ intersects in $D$ , and $B$ and $D$ are on different sides for line $AC$ . $M$ - midpoint of $AC$ . Prove, that $AE,CF,DM$ intersects at one point.