# Mock AIME 1 2006-2007 Problems/Problem 7

## Problem

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 ).

## Solution

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

Therefore .

## Alternate Solution

As above, use Power of a Point to compute and . Since triangles and share the same height, . Similarly, . Using Menelaus's Theorem on points on the sides of triangle , we see that Let . Using Ceva's Theorem on points lying on the sides of triangle , we find that Then, using Menelaus's Theorem on the points on the sides of triangle , we see that Thus, so that