KöMaL 2013 A591

There is given a convex quadrilateral ABCD and some points $P$, $Q$, $R$ and $S$on the line segments $AB$, $BC$, $CD$ and $DA$, respectively. The line segments $PR$and $QS$ meet at $T$. Suppose that each of the the quadrilaterals APTS , BQTP , CRTQ and DSTR have an inscribed circle. Prove that the quadrilateral ABCD also has an inscribed circle.