Τρίτη, 17 Οκτωβρίου 2017

2003 IMO Shortlist G7

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$.  The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that $ \frac{s}{2}< t \le \frac{s}{2}  + \Big(1 -\frac{\sqrt{3}}{2}\Big) r $.

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