ZA-SAMO 2003 Problem 6

In $\Delta ABC$ , the sum of the sides is $2s$ and the radius of the incircle is $r$ . Three semicircles with diameters $AB$ , $BC$ and $CA$ are drawn on the outside of $ABC$ . A circle with radius $t$ touches all three semicircles. Prove that $$\frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r.$$