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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 of these three semicircles has radius t. Prove that
\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r.
Alternative formulation. In a triangle ABC, construct circles with diameters BC, CA, and AB, respectively. Construct a circle w externally tangent to these three circles. Let the radius of this circle w be t.
Prove: \frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r, where r is the inradius and s is the semiperimeter of triangle ABC.

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