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2022 PUMaC Geometry A5 / B7

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September 10, 2023
geometry

Problem Statement

Let △ABC\vartriangle ABC be a triangle with AB=5AB = 5, BC=8BC = 8, and, CA=7CA = 7. Let the center of the AA-excircle be OO, and let the AA-excircle touch lines BCBC, CACA, and,AB AB at points X,YX, Y , and, ZZ, respectively. Let h1h_1, h2h_2, and, h3h_3 denote the distances from OO to lines XYXY , YZY Z, and, ZX, respectively. If h12+h22+h32h^2_1+ h^2_2+ h^2_3 can be written as mn\frac{m}{n} for relatively prime positive integers m,nm, n, find m+nm + n.
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