Difference between revisions of "2021 JMC 10 Problems/Problem 21"
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Revision as of 15:55, 1 April 2021
Problem
Two identical circles and with radius have centers that are units apart. Two externally tangent circles and of radius and respectively are each internally tangent to both and . If , what is ?
Solution
Let and be the centers of and respectively. Let be the radius of and and be the distance between and . Note that the centers of and , say and respectively, lie on a line that is both perpendicular to and equidistant from and .
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Because , we have that is the length of the -altitude of . We have , , and , so 's perimeter is . Thus, by Heron's Formula . Substituting known values, we have whence .
Remark: In this specific case, is actually a right triangle with lengths in the ratio , which is why the diagram has one of the centers lying on .