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a wild symmedian appears

Source: 2019 AIME I #15

March 14, 2019
AIMEAIME I2019 AIME Igeometrysymmedianradical axispower of a point

Problem Statement

Let AB‾\overline{AB} be a chord of a circle ω\omega, and let PP be a point on the chord AB‾\overline{AB}. Circle ω1\omega_1 passes through AA and PP and is internally tangent to ω\omega. Circle ω2\omega_2 passes through BB and PP and is internally tangent to ω\omega. Circles ω1\omega_1 and ω2\omega_2 intersect at points PP and QQ. Line PQPQ intersects ω\omega at XX and YY. Assume that AP=5AP=5, PB=3PB=3, XY=11XY=11, and PQ2=mnPQ^2 = \tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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