Question 152
Let $ABCD$ be a cyclic quadrilateral. Let $O$ be the circumcenter of the quadrilateral $ABCD$. The diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R$ and $S$ be the circumcenters of triangles $AGB, BGC, CGD$ and $DGA$ respectively. The lines $PR$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $G$ and $O$.