Question 11
Triangle $ABC$ has side lengths $AB = 7,$ $BC = 8,$ and $CA = 9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$