BALTIC-WAY 2012 Problem 15

The circumcentre $O$ of a given cyclic quadrilateral $A B C D$ lies inside the quadrilateral but not on the diagonal $A C$. The diagonals of the quadrilateral intersect at $I$.

The circumcircle of the triangle $A O I$ meets the sides $A D$ and $A B$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $C O I$ meets the sides $C B$ and $C D$ at points $R$ and $S$, respectively. Prove that $P Q R S$ is a parallelogram.