Question 1
Prove for each triangle $ABC$ the inequality
$$\frac14\lt \frac{IA\cdot IB\cdot IC}{l_A l_B l_C}\le\frac8{27},$$
where $I$ is the incenter and $l_A,l_B,l_C$ are the lengths of the angle bisectors of $ABC$.
$$\frac14\lt \frac{IA\cdot IB\cdot IC}{l_A l_B l_C}\le\frac8{27},$$
where $I$ is the incenter and $l_A,l_B,l_C$ are the lengths of the angle bisectors of $ABC$.