Find The Length

Geometry Level 5

Let ABC\triangle ABC be a triangle with AB=6,BC=7,CA=8.AB= 6, BC = 7, CA=8. Let P,QP,Q be the feet of perpendiculars from B,CB,C on CA,ABCA,AB respectively. Let BPBP and CQCQ meet at H,H, and let PQPQ and BCBC meet at X.X. Line AXAX meets the circumcircle of ABC\triangle ABC at Y,Y, where YA.Y \neq A. Lines HYHY and BCBC intersect at J.J. Given that AJ2=mnAJ^2 = \dfrac{m}{n} for some coprime positive integers m,n,m,n, find m+n.m+n.

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