Problem

Source: Ukraine TST 2011 p9

Tags: geometry, circumcircle, equal angles, tangent circles, Tangents, cyclic quadrilateral



Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.