Problem 1701

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 $BCD$.
(5 pont)