Question 5
For the vertex $A$ of $\triangle ABC$, define $\ell_A$ by the distance between the feet of perpendiculars drawn to the sides $AB$ and $AC$ from the intersection of the angle bisector of $\angle A$ and the side $BC$. Similarly, define $\ell_B$ and $\ell_C$ for the vertices $B$ and $C$, respectively. Prove the inequality
$$
\frac{\ell_A \ell_B \ell_C}{\ell^3} \le \frac{1}{64}
$$
where $\ell$ is the perimeter of $\triangle ABC$.
$$
\frac{\ell_A \ell_B \ell_C}{\ell^3} \le \frac{1}{64}
$$
where $\ell$ is the perimeter of $\triangle ABC$.