Let $X$ be the intersection of lines perpendicular to $BD$ and $AC$ passing through $A$ and $B$. Similarly we define $Y$ for $B,C$, $Z$ for $C,D$ and $T$ for $A,D$ and we get parallelogram $XYZT$. Consider homothety at $G$ and scale $2$. It's sufficient to prove that $O$ is the midpoint of $XZ, YT$. Let $B', D'$ be respectively second intersection of $XY, ZT$ with circle $ABCD$. Then $BB'D'D$ is a rectangle so $O$ lies in equal distance between lines $XY, ZT$. Analogously $O$ lies in equal distance between lines $XZ, YT$ which proves the claim.

Prove that $ P,S,O,A-concyclic $