Problem

Source: Iran 2004

Tags: geometry, parallelogram, rectangle, geometric transformation, homothety, circumcircle, cyclic quadrilateral



Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .