2023 SSMO Relay Round 3 Problems/Problem 1

Problem

In triangle $ABC$ with $AB=13,AC=14,BC=15$ , circles $\omega_1,\omega_2,$ and $\omega_3$ are drawn, centered at $A,B$ and $,C$ , respectively. Each of the three circles are externally tangent to the two other circles. If the radius of a circle $\omega$ such that $\omega$ is internally tangent to $\omega_1,\omega_2,$ and $\omega_3$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n$ , find $m+n.$

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2023 SSMO Relay Round 3 Problems/Problem 1

Problem

Solution