2007 AIME II Problem 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.