Problem:
Points and lie on a circle centered at , and . A second circle is internally tangent to the first and tangent to both and . What is the ratio of the area of the smaller circle to that of the larger circle?
Answer Choices:
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Solution:
Let and be the radii of the smaller and larger circles, respectively. Let be the center of the smaller circle, let be the radius of the larger circle that contains , and let be the point of tangency of the smaller circle to . Then , and because is a triangle, . Thus , so . The ratio of the areas is
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The problems on this page are the property of the MAA's American Mathematics Competitions