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 $$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$