Question 36
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and tangent to $AB, AC$ at $E, F$, respectively. The tangents from $M$ to $\omega$ meet this circle at $P, Q$ such that $P$ and $B$ lie on the same side of $AM$. Lines $PM, BF$ cut each other at $X$, and $Y$ is the intersection of $QM, CE$. If $2PM = BC$, prove that $XY$ is tangent to $\omega$.