Question 9
In right triangle $ABC$ with right angle $C,$ $CA = 30$ and $CB = 16.$ Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B.$ Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to the hypotenuse and to the extension of leg $CA,$ the circle with center $O_2$ is tangent to the hypotenuse and to the extension of leg $CB,$ and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$