AT-OEMO-BEG 2022 Problem 3

A semicircle is erected over the segment $AB$ with center $M$ . Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$ . The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$ . Prove that $PM = PS$ holds.

(Karl Czakler)