Mock AIME I 2015 Problems/Problem 3
Let be points in the plane such that , , and . Semicircles with diameters and intersect at a point with . Find the length of line segment .
First note that is a 3-4-5 scaled right triangle with and is a 20-21-29 right triangle with since and are inscribed in semicircles, and thus right angles. Also, since and are right, points B, P and C are collinear and
Solution by D. Adrian Tanner
First note that is a 3-4-5 right triangle and is a 20-21-29 right triangle. Thus we have and . Similarly, we have and . Applying the Law of Cosines on gives us We also know that Hence, we have