KöMaL 2015 A636

There is given a convex quadrilateral $ABCD$ and a point $P$ in the interior of the triangle $BCD$ in such a way that the quadrilateral $ABPD$ has an inscribed circle, and the three inscribed circles of the quadrilateral $ABPD$ , the triangle $BCP$ and the triangle $CDP$ , respectively, are pairwise tangent to each other. Denote by $Q$ and $R$ the points tangency on the line segments $BP$ and $DP$ , respectively. Let the lines $BP$ and $AR$ meet at $S$ , let the lines $DP$ and $AQ$ meet at $T$ , and let the lines $BT$ and $DS$ meet at $U$ . Show that the line $CU$ bisects the angle $BCD$ .