Three Chords & The Truth

Geometry Level 4

Segments \(\overline{PA}\), \(\overline{PB}\), and \(\overline{PC}\) are all chords of the same circle, \(\Gamma\), such that \(PA=5\), \(PB=8\), and \(PC=3\). In addition, \(\angle APB \cong \angle BPC.\) The area of \(\Gamma\) can be written as \(\frac{a}{b}\pi\), where \(a\) and \(b\) are positive, coprime integers. Find \(a+b\).

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