Find The Length

Geometry Level 5

Let \(\triangle ABC\) be a triangle with \(AB= 6, BC = 7, CA=8.\) Let \(P,Q\) be the feet of perpendiculars from \(B,C\) on \(CA,AB\) respectively. Let \(BP\) and \(CQ\) meet at \(H,\) and let \(PQ\) and \(BC\) meet at \(X.\) Line \(AX\) meets the circumcircle of \(\triangle ABC\) at \(Y,\) where \(Y \neq A.\) Lines \(HY\) and \(BC\) intersect at \(J.\) Given that \(AJ^2 = \dfrac{m}{n}\) for some coprime positive integers \(m,n,\) find \(m+n.\)

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