Question 10
Let $\triangle ABC$ have side lengths $AB = 5,$ $BC = 9,$ and $CA = 10.$ The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D,$ and $\overline{AD}$ intersects the circumcircle at $P \ne A.$ The length of $AP$ is equal to $\frac{m}{n},$ where $m$ and $n$ are relatively prime integers. Find $m + n.$