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Bisectors, perpendicularity and circles

Source: Pan-American Girls’ Mathematical Olympiad 2022, Problem 3

October 27, 2022
geometryPAGMO

Problem Statement

Let ABCABC be an acute triangle with AB<ACAB< AC. Denote by PP and QQ points on the segment BCBC such that ∠BAP=∠CAQ<∠BAC2\angle BAP = \angle CAQ < \frac{\angle BAC}{2}. B1B_1 is a point on segment ACAC. BB1BB_1 intersects APAP and AQAQ at P1P_1 and Q1Q_1, respectively. The angle bisectors of ∠BAC\angle BAC and ∠CBB1\angle CBB_1 intersect at MM. If PQ1⊥ACPQ_1\perp AC and QP1⊥ABQP_1\perp AB, prove that AQ1MPBAQ_1MPB is cyclic.
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