# Mock Geometry AIME 2011 Problems/Problem 11

## Problem

is on a semicircle with diameter and center Circle radius is tangent to and arc and circle radius is tangent to and arc . It is known that . The ratio can be expressed where are relatively prime positive integers. Find

## Solution

Let the circle with radius have center and the circle with radius have center . Let the projections of and onto be and , respectively.

is equidistant from and , so it is on the angle bisector of .

We're given that , so . Now, we have , which is positive because .

We therefore also have . Now, , and we see that the radius of the large semicircle is .

Similarly, is the angle bisector of . Now, , and so , again positive because , and .

Now, . The radius of the large semicircle is thus .

Since the radii of the large semicircle are equal, we have , and so , and .