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right triangle, 2 circles given, angle bisector wanted

Source: 2014 Austrian Federal Competition For Advanced Students, Part 1 p4

February 2, 2020
geometryangle bisectorright triangle

Problem Statement

We are given a right-angled triangle MNPMNP with right angle in PP. Let kMk_M be the circle with center MM and radius MPMP, and let kNk_N be the circle with center NN and radius NPNP. Let AA and BB be the common points of kMk_M and the line MNMN, and let CC and DD be the common points of kNk_N and the line MNMN with with CC between AA and BB. Prove that the line PCPC bisects the angle ∠APB\angle APB.
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