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0322 geo inequality 3rd edition Round 2 p2

Source:

May 9, 2021
inequalities3rd edition

Problem Statement

Let ABCABC be a triangle with semiperimeter ss and inradius rr. The semicircles with diameters BC,CA,ABBC, CA, AB are drawn on the outside of the triangle ABCABC. The circle tangent to all three semicircles has radius tt. Prove that s2<t≤s2+(1−32)r.\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.
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