IR-TST 2018 Problem 3

In triangle $ABC$ let $M$ be the midpoint of $BC$ . Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$ , respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$ . Let $X \equiv PM \cap BF$ and $Y \equiv QM \cap CE$ . If $2PM=BC$ prove that $XY$ is tangent to $\omega$ .

Proposed by Iman Maghsoudi