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IMO ShortList 2003, geometry problem 7

Source: IMO ShortList 2003, geometry problem 7

June 4, 2004
geometryperimetercircumcirclegeometric inequalityTriangleIMO Shortlist

Problem Statement

Let ABCABC be a triangle with semiperimeter ss and inradius rr. The semicircles with diameters BCBC, CACA, ABAB are drawn on the outside of the triangle ABCABC. The circle tangent to all of these three semicircles has radius tt. Prove that s2<t≤s2+(1−32)r.\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. Alternative formulation. In a triangle ABCABC, construct circles with diameters BCBC, CACA, and ABAB, respectively. Construct a circle ww externally tangent to these three circles. Let the radius of this circle ww be tt. Prove: s2<t≤s2+12(2−3)r\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r, where rr is the inradius and ss is the semiperimeter of triangle ABCABC.
Proposed by Dirk Laurie, South Africa
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