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exists circle tangent to 4 circumcircles, #, <APB+<CPD=<BPC+<APD

Source: 2017 All-Ukrainian Correspondence MO, grades 5-12 p11

April 29, 2021
geometrytangent circlesparallelogramUkraine Correspondence

Problem Statement

Inside the parallelogram ABCDABCD, choose a point PP such that ∠APB+∠CPD=∠BPC+∠APD\angle APB+ \angle CPD= \angle BPC+ \angle APD. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles APBAPB, BPCBPC, CPDCPD and APDAPD.
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