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)