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A circle of radius s containing all three points [Czech-Polish-Slovak 2018]

Source: Czech-Polish-Slovak Match 2018, Problem 4

July 2, 2018
Geometric Inequalitiesgeometryinequalities

Problem Statement

Let ABCABC be an acute triangle with the perimeter of 2s2s. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers A,BA, B, and CC, respectively. Prove that there exists a circle with the radius of ss which contains all the three circles.
Proposed by Josef Tkadlec, Czechia