1997 OIM Problems/Problem 2

Problem

With the center at the incenter $I$ of a triangle $ABC$ , a circle is drawn that cuts each of the three sides of the triangle at two points: the segment $BC$ at $D$ and $P$ ( $D$ being the closest to $B$ ); the the segment $CA$ in $E$ and $Q$ ( $E$ being the closest to $C$ ), and the segment $AB$ in $F$ and $R$ ( $F$ being the closest to $A$ ). Let $S$ be the point of intersection of the diagonals of the quadrilateral $EQFR$ . Let $T$ be the point of intersection of the diagonals of the $FRDP$ quadrilateral. Let $U$ be the point of intersection of the diagonals of the quadrilateral $DPEQ$ . Show that the circles circumscribed by the triangles $FRT$ , $DPU$ and $EQS$ have a single common point.