TH-TMO 2017 Problem 2

A cyclic quadrilateral $ABCD$ has circumcenter $O$ , its diagonals $AC$ and $BD$ intersect at $G$ . Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$ . Show that $M$ is the midpoint of $OG$ .