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2022 PUMaC Geometry A7

Source:

September 10, 2023
geometry

Problem Statement

Let △ABC\vartriangle ABC be a triangle with BC=7BC = 7, CA=6CA = 6, and, AB=5AB = 5. Let II be the incenter of △ABC\vartriangle ABC. Let the incircle of △ABC\vartriangle ABC touch sides BCBC, CACA, and ABAB at points D,ED,E, and FF. Let the circumcircle of △AEF\vartriangle AEF meet the circumcircle of △ABC\vartriangle ABC for a second time at point X≠AX\ne A. Let PP denote the intersection of XIXI and EFEF. If the product XP⋅IPXP \cdot IP can be written as mn\frac{m}{n} for relatively prime positive integers m,nm, n, find m+nm + n.
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