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Diameters, collinearity iff bisector

Source: Pan-American Girls' Mathematical Olympiad 2022, P4

October 30, 2022
geometryPAGMO

Problem Statement

Let ABCABC be a triangle, with AB≠ACAB\neq AC. Let O1O_1 and O2O_2 denote the centers of circles ω1\omega_1 and ω2\omega_2 with diameters ABAB and BCBC, respectively. A point PP on segment BCBC is chosen such that APAP intersects ω1\omega_1 in point QQ, with Q≠AQ\neq A. Prove that O1O_1, O2O_2, and QQ are collinear if and only if APAP is the angle bisector of ∠BAC\angle BAC.
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