KöMaL 2007 B3991

The isosceles triangle ABC has a right angle at $C$. Let $P$ be an arbitrary point of side $BC$, and let $G$ denote the orthogonal projection of the point $C$ onto $AP$. Let $H$ be the point of the line segment $AP$ for which $AH$=$CG$. Find the angle subtended by the line segment $GH$ at the midpoint of $AB$.