¶ 2025 AMC 12B Problem 25
¶ Problem:
Three concentric circles have radii . An equilateral triangle with side length has one vertex on each circle. What is
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¶ Solution:
Suppose , , and are the vertices of the equilateral triangle on the circles of radius , , and , and suppose that point is the center of the concentric circles, so that our diagram is as follows:
Since , we must have that lies on the circumcircle of , by the converse of Ptolemy's theorem. This is depicted below.
Hence, . Since and , we have that:
by Law of Cosines, so the answer is .
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