Question 138
Let $ABC$ be a triangle and let $D$ be a point on $BC$, let $E$ be a point on $AC$, let $F$ be a point on $AB$ such that $\angle BFD = \angle AFE$ and $\angle AEF = \angle CED$. Let $M$ and $N$ be points on the line $EF$ such that $EN = FD$ and $FM = ED$, where the points $M, F, E, N$ are positioned in this order. The perpendicular bisector of the segment $DN$ meets the line $AB$ at point $P$ and the perpendicular bisector of the segment $DM$ meets the line $AC$ at point $Q$. Prove that $PQ$ is perpendicular to $MN$.