**A. 591.** 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.

(5 points)