If $a,b,c$ are positive reals satisfying $a+b+c \geq abc$, prove that: $a^2+b^2+c^2 \geq \sqrt 3abc$

Question

If $a,b,c$ are positive reals satisfying $a+b+c \geq abc$, prove that: $a^2+b^2+c^2 \geq \sqrt 3abc$

I've been trying to solve the above question since a long time but I'm not able to. I'm not able to understand how to use this condition (I've generally dealt with inequalities having conditions as a sum = constant). Would someone please help me to solve this question? Thanks in advance!

Answer 1

Because $$a^2+b^2+c^2\geq\sqrt{3abc(a+b+c)}\geq\sqrt3abc.$$