Question 9
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 $AP=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$