MathNet 2000 — Math Olympiad Vault Question 38

Given an isosceles triangle $A B C$ with $\angle A=90^\circ$. Let $M$ be the midpoint of $A B$. The line passing through $A$ and perpendicular to $C M$ intersects the side $B C$ at $P$. Prove that $\angle A M C=\angle B M P$.