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2022 W CHMMC Tiebreaker 4 - expected value of (PXQ)

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March 2, 2024
geometryCHMMC

Problem Statement

Let ABCABC be a triangle with AB=4AB = 4, BC=5BC = 5, CA=6CA = 6. Triangles APBAPB and CQACQA are erected outside ABCABC such that AP=PBAP=PB, AP‾⊥PB‾\overline{AP}\perp \overline{PB} and CQ=QACQ=QA, CQ‾⊥QA‾\overline{CQ}\perp \overline{QA}. Pick a point XX uniformly at random from segment BC‾\overline{BC}. What is the expected value of the area of triangle PXQPXQ?
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