Question 41
Let $E$ be an interior point in the convex quadrilateral $ABCD$. Let $F$, $G$, $H$, and $I$ be points opposite the quadrilateral with respect to the lines $AB$, $BC$, $CD$, and $DA$, respectively, such that $\triangle ABF \sim \triangle DCE$, $\triangle BCG \sim \triangle ADE$, $\triangle CDH \sim \triangle BAE$, and $\triangle DAI \sim \triangle CBE$. 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
$$
EF \cdot CD = EG \cdot DA = EH \cdot AB = EI \cdot BC.
$$
$$
EF \cdot CD = EG \cdot DA = EH \cdot AB = EI \cdot BC.
$$