ELMO 2020 Problem 4

Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$ . Let $M$ be the midpoint of side $BC$ , and let $D'$ be the reflection of $D$ over $M$ . Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$ . Prove that $\angle MHG = 90^\circ$ .

Proposed by Daniel Hu.