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3 circles given, collinearity wanted

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 2014 p5

February 11, 2020
circlescollineargeometry

Problem Statement

Given is the acute triangle ABCABC. Let us denote kk a circle with diameter ABAB. Another circle, tangent to ABAB at point AA and passing through point CC intersects the circle kk at point P,P≠AP, P \ne A. Another circle which touches AB at point BB and passes point CC, intersects the circle kk at point Q,Q≠BQ, Q \ne B. Prove that the intersection of the line AQAQ and BPBP lies on one of the sides of angle ACBACB.
(Peter Novotný)
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