CONO-SUR 2022 Problem 2

Given is a triangle $ABC$ with incircle $\omega$ , tangent to $BC, CA, AB$ at $D, E, F$ . The perpendicular from $B$ to $BC$ meets $EF$ at $M$ , and the perpendicular from $C$ to $BC$ meets $EF$ at $N$ . Let $DM$ and $DN$ meet $\omega$ at $P$ and $Q$ . Prove that $DP=DQ$ .