2009 AIME II Problems/Problem 10

Four lighthouses are located at points , , , and . The lighthouse at is kilometers from the lighthouse at , the lighthouse at is kilometers from the lighthouse at , and the lighthouse at is kilometers from the lighthouse at . To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. The number of kilometers from to is given by , where , , and are relatively prime positive integers, and is not divisible by the square of any prime. Find + + .

Solution

Let be the intersection of and . By the Angle Bisector Theorem, = , so = and = , and + = = , so = , and = . Let be the altitude from to . It can be seen that triangle is similar to triangle , and triangle is similar to triangle . If = , then = , = , and = . Since + = = , = , and = . The answer is + + = .