In triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent to line $BC$ at point $C$. If $r = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $100m + n$.

Let's think step by step and output the final answer within \boxed{}.
user