Question 119
Let $ABC$ be a triangle with $\angle A = 90^\circ$ and $AB = 2AC$. A point $M$ is chosen on the side $AC$. Let $P$ be the foot of the perpendicular from $A$ to the line $BM$, and let $Q$ be the foot of the perpendicular from $C$ to the line $BM$. Prove that $4PQ + 2QC = BP$.