Question 10
Four lighthouses are located at points $A,$ $B,$ $C,$ and $D.$ The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B,$ the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C,$ and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C.$ To an observer at $A,$ the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C,$ the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac{p\sqrt{r}}{q},$ where $p,$ $q,$ and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p + q + r.$