2009 AIME II Problem 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.