How to prove that $2 \min(|a−b|,|b−c|,|c−a|)$ ≤ $R$ How to prove that $2 \min(|a−b|,|b−c|,|c−a|)$ ≤ $R$

I know that min is for $\sqrt{7}-1,\sqrt{7},\sqrt{7}+1$, but how to prove this?

We may assume without loss of generality that the side lengths are given by $2x-y,2x,2x+y$ with $y\leq x$ in order to fulfill the triangle inequality. With such assumptions, the area of $ABC$, by Heron's formula, is given by:
$$ \Delta = \frac{\sqrt{3}}{2}\sqrt{x^2(2x-2y)(2x+2y)} $$
and the circumradius is given by:
$$ R = \frac{abc}{4R} = \frac{1}{2\sqrt{3}}\sqrt{\frac{(2x-y)^2(2x+y)^2}{(2x-2y)(2x+2y)}}=\frac{1}{4\sqrt{3}}\sqrt{\frac{(4x^2-y^2)^2}{x^2-y^2}} $$
so the problem boils down to studying a quadratic form in $x^2$ and $y^2$, pretty easy.
