ELMO 2011 Problem 1

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$ , $BC$ , $CD$ , $DA$ , respectively, and let $P$ be the intersection of $EG$ and $FH$ . Given that quadrilaterals $HAEP$ , $EBFP$ , $FCGP$ , $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.

Evan O'Dorney.