2010 AIME II Problems/Problem 15
Problem 15
In triangle , , , and . Points and lie on with and . Points and lie on B with and . Let be the point, other than , of intersection of the circumcircles of and . Ray meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .
Solution
Let . since . Since quadrilateral is cyclic, and , yielding and . Multiplying these together yields .
. Also, is the center of spiral similarity of segments and , so . Therefore, , which can easily be computed by the angle bisector theorem to be . It follows that , giving us an answer of .
Note: Spiral similarities may sound complex, but they're really not. The fact that is really just a result of simple angle chasing.
Source: [1] by Zhero