Trapezoid \(ABCD^{}_{}\) has sides \(AB=92^{}_{}\), \(BC=50^{}_{}\), \(CD=19^{}_{}\), and \(AD=70^{}_{}\), with \(AB^{}_{}\) parallel to \(CD^{}_{}\). A circle with center \(P^{}_{}\) on \(AB^{}_{}\) is drawn tangent to \(BC^{}_{}\) and \(AD^{}_{}\). Given that \(AP^{}_{}=\frac mn\), where \(m^{}_{}\) and \(n^{}_{}\) are relatively prime positive integers, find \(m+n^{}_{}\).

(第一十届AIME1992 第9题)