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tangent circles of BDX, CEX iff ABxAD<=4R^2 2020 PUMaC Individual Finals B3

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January 1, 2022
geometrytangent circlesgeometric inequality

Problem Statement

Let ABCABC be a triangle and let the points D,ED, E be on the rays ABAB, ACAC such that BCEDBCED is cyclic. Prove that the following two statements are equivalent: ∙\bullet There is a point XX on the circumcircle of ABCABC such that BDXBDX, CEXCEX are tangent to each other. ∙\bullet AB⋅AD≤4R2AB \cdot AD \le 4R^2, where RR is the radius of the circumcircle of ABCABC.
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