Question Number 173426 by Shrinava last updated on 11/Jul/22
$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship} \\ $$$$\mathrm{holds}: \\ $$$$\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \right)\left(\mathrm{h}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{3}} +\mathrm{h}_{\boldsymbol{\mathrm{b}}} ^{\mathrm{3}} +\mathrm{h}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{3}} \right)\:\geqslant\:\mathrm{5832}\:\sqrt{\mathrm{3}}\:\mathrm{r}^{\mathrm{6}} \\ $$