2017/031) Prove that A triangle ABC is equilateral if and only if $\tan\, A + \tan\, B + \tan\, C = 3\sqrt{3}$

First let us prove the if part
if ABC is equilateral we have $\tan\, A + \tan\, B + \tan\, C = \tan\, 60^{\circ}+\tan\, 60^{\circ} + \tan\, 60^{\circ} = 3\tan\, 60^{\circ} =\sqrt{3}$
Now for the other part
we have using AM GM inequality(for all 3 positive) 
$\frac{\tan\, A + \tan\, B + \tan\, C}{3} >=\sqrt[3]{\tan\, A\,  \tan\, B\, \tan\, C}$
or $\tan\, A + \tan\, B + \tan\, C >=3\sqrt[3]{\tan\, A\,  \tan\, B\, \tan\, C}$
and these are equal if $\tan\, A = \tan\, B = \tan\, C$ or $A=B=C$ and at this  we have the sum = $3\sqrt[3]3$