Let $M\;$ be a point in the interior of $\Delta ABC,\;$ and $AA_0,\;$ $BB_0,\;$ $CC_0\;$ the cevians through $M.\;$ Then by Gergonne's Theorem, and, applying the AM-GM inequality,

$\displaystyle 1=\frac{MA_0}{AA_0}+\frac{MB_0}{BB_0}+\frac{MC_0}{CC_0}\ge 3\sqrt[3]{\frac{MA_0}{AA_0}\cdot\frac{MB_0}{BB_0}\cdot\frac{MC_0}{CC_0}}.$

In particular, $\displaystyle 1\ge 27\frac{IA'}{\ell_a}\cdot\frac{IB'}{\ell_b}\cdot\frac{IC'}{\ell_c}$ and $\displaystyle 1\ge 27\frac{HA''}{h_a}\cdot\frac{HB''}{h_b}\cdot\frac{HC''}{h_c}$ whose product gives the required result.

The inequality and the solution have been kindly communicated to me by Dan Sitaru.