Question 25
In triangle $ABC$, $\angle B = 90^\circ$, $AB \gt BC$, and $P$ is the point such that $BP = BC$ and $\angle APB = 90^\circ$, where $P$ and $C$ lie on the same side of $AB$. Let $Q$ be the point on $AB$ such that $AP = AQ$, and let $M$ be the midpoint of $QC$. Prove that the line through $M$ parallel to $AP$ passes through the midpoint of $AB$.