BALTIC-WAY 2011 Problem 13

Let $E$ be an interior point of the convex quadrilateral $ABCD$ . Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $\triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$ , respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then $$EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.$$