2009 AIME II Problems/Problem 10

Four lighthouses are located at points A, B, C, and D. The lighthouse A is 5 km away from the lighthouse at B, the lighthouse at B is 12 km from the lighthouse at C, and the lighthouse at A is 13 km 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 are equal, and the angle determined by the lights at D and B are equal. The number of km from A to D is given by (p*sqrt (q))/r 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.