AIME 2015 Problem 11

Geometry Level 4

The circumcircle of acute \(\Delta ABC\) has center \(O\). The line passing through point \(O\) perpendicular to \(OB\) intersects lines \(AB\) and \(BC\) and \(P\) and \(Q\), respectively. Also \(AB=5\), \(BC=4\), \(BQ=4.5\), and \(BP=\dfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).


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