Question 142
Let $\triangle ABC$ be a right triangle with $\angle B = 90^\circ$ and $|AB| \gt |BC|$; and let $\Gamma$ be the semicircle with diameter $AB$ on the side of $AB$ where $C$ also lies. Let $P$ be the point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be the point on segment $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.