Question 15
Four circles $\omega,$ $\omega_A,$ $\omega_B,$ and $\omega_C$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_A$ is tangent to sides $AB$ and $AC,$ $\omega_B$ to $BC$ and $BA,$ $\omega_C$ to $CA$ and $CB,$ and $\omega$ is externally tangent to $\omega_A,$ $\omega_B,$ and $\omega_C.$ If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$