2009 AIME II Problems/Problem 5
Equilateral triangle is inscribed in circle , which has radius . Circle with radius is internally tangent to circle at one vertex of . Circles and , both with radius , are internally tangent to circle at the other two vertices of . Circles , , and are all externally tangent to circle , which has radius , where and are relatively prime positive integers. Find .
Let be the intersection of the circles with centers and , and be the intersection of the circles with centers and . Since the radius of is , . Assume = . Then and are radii of circle and have length . , and angle degrees because we are given that triangle is equilateral. Using the Law of Cosines on triangle , we obtain
The and the terms cancel out:
. The radius of circle is , so the answer is .
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