Question 19
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
$$ \frac{s}{2} \lt t\leq\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. $$
$$ \frac{s}{2} \lt t\leq\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. $$