MathNet 2013 Question 9

Let $ABC$ be a triangle and let points $M$ and $N$ on the sides $AB$ and $AC$ respectively such that $\angle ABC \equiv \angle ANM$. Let $D$ be the mirror image of point $A$ across $B$, and let $P$ and $Q$ be the midpoints of the line segments $MN$ and $CD$ respectively. Show that points $A$, $P$ and $Q$ are collinear if and only if $AC = AB\sqrt{2}$.
Petru Braica