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Source: South Africa 2003

October 14, 2005
geometrygeometry solved

Problem Statement

In ΔABC\Delta ABC, the sum of the sides is 2s2s and the radius of the incircle is rr. Three semicircles with diameters ABAB, BCBC and CACA are drawn on the outside of ABCABC. A circle with radius tt touches all three semicircles. Prove that s2<t≤s2+(1−32)r. \frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r.
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