Mock AIME 1 2006-2007 Problems/Problem 7
Let have and . Point is such that and . Construct point on segment such that . and are extended to meet at . If where and are positive integers, find (note: denotes the area of ).
We can immediately see that quadrilateral is cyclic, since . We then have, from Power of a Point, that . In other words, . is then 2, and is 1. We can now use Menelaus on line with respect to triangle :
This shows that .
Now let , for some real . Therefore , and . Similarly, and . The desired ratio is then