Question 2
Let $ABC$ be a right triangle with $\angle C=90^\circ$, and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of $\triangle ACD$ and $F$ the centroid of $\triangle BCD$. The point $P$ satisfies $\angle CEP=90^\circ$ and $|CP|=|AP|$, while the point $Q$ satisfies $\angle CFQ=90^\circ$ and $|CQ|=|BQ|$.
Prove that $PQ$ passes through the centroid of $\triangle ABC$.
Prove that $PQ$ passes through the centroid of $\triangle ABC$.