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Easy right-angled triangle problem

Source: SMO open 2024 Q1

August 3, 2024
geometry

Problem Statement

In triangle ABCABC, ∠B=90∘\angle B=90^\circ, AB>BCAB>BC, and PP is the point such that BP=BCBP=BC and ∠APB=90∘\angle APB=90^\circ, where PP and CC lie on the same side of ABAB. Let QQ be the point on ABAB such that AP=AQAP=AQ, and let MM be the midpoint of QCQC. Prove that the line through MM parallel to APAP passes through the midpoint of ABAB.
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