Mock Geometry AIME 2011 Problems/Problem 7
In trapezoid and There is a point on side such that the circumcircle of triangle is tangent to If can be expressed in the form where are positive integers and are relatively prime. Find
Let be the foot of the perpendicular from to . Then . By the Pythagorean Theorem on , . Solving yields , and so .
Let be the intersection of lines and . by similarity. Then , or , and so . Also, , or , and so . Therefore, . By the Power of a Point of , . Then .
From the Law of Cosines of , . Solving yields . Similarly, from the Law of Cosines of , . Solving yields .
Then , and so .