9. In trapezoid $A B C D$, $A B \| C D$, and $A B$ $=92, B C=50, C D=19, A D=70$. A circle with center $P$ on $A B$ is tangent to sides $B C$ and $A D$. If $A P=\frac{m}{n}$, where $m, n$ are coprime positive integers. Find $m+n_{0} \quad$
9. As shown in the figure, extend $A D$ and $B C$ to intersect at point $Q$. Since the distances from $P$ to $A Q$ and $B Q$ are equal, $P$ lies on the angle bisector of $\angle A Q B$, thus we have
$$
\frac{A P}{B P}=\frac{A Q}{B Q}
$$

Since $A B \parallel C D$, we have
$$
\frac{A Q}{B Q}=\frac{A D}{B C}=\frac{7}{5}.
$$

From (1) and (2), we get $\frac{A P}{B P}=\frac{7}{5}$.
$$
\because \quad A P+P B=92.
$$
Substituting (4) into (3), we get $A P=\frac{161}{3}$. Therefore,
$$
m+n=161+3=164
$$