The Intersections of Extended Cevians with Three Circumcircles of Subtriangles

Let P be a point in the interior of the triangle ABC. Draw the three circumscribed circles for the triangles APB, APC, and BPC. Let X, Y, and Z be the intersections (other than P) of the extensions of AP, BP, and CP with the circles opposite A, B, and C. Then:
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See Ukrainian Journal Contest, Problem 326, by Olexandr Manzjuk.
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