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HMMT Geometry 2019/7: Do you remember your triangle geometry?

Source:

February 17, 2019
HMMTgeometry

Problem Statement

Let ABCABC be a triangle with AB=13AB = 13, BC=14BC = 14, CA=15CA = 15. Let HH be the orthocenter of ABCABC. Find the radius of the circle with nonzero radius tangent to the circumcircles of AHBAHB, BHCBHC, CHACHA.
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