2015 Mock AIME I Problem 3: Intersection of Circles

Level pending

Let \(A,B,C\) be points in the plane such that \(AB=25\), \(AC=29\), and \(\angle BAC<90^\circ\). Semicircles with diameters \(\overline{AB}\) and \(\overline{AC}\) intersect at a point \(P\) with \(AP=20\). Find the maximum possible length of line segment \(\overline{BC}\).

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