R and r together

Let the side of the equilateral triangle be $a.$
We know that the formula for circum-radius $R=\dfrac{abc}{4\delta}$ and in-radius $r=\dfrac{2\delta}{(a+b+c)}$
where $a,b,c$ are the sides and $\delta$ is the area of the triangle $=\dfrac{a^2\sqrt3}{4}$ by equilateral triangle area formula.
$R=\dfrac{\cancel{a}\times\cancel{a}\times a}{\cancel{4}\times\frac{\cancel{a^2}\sqrt3}{\cancel{4}}}=\dfrac{a}{\sqrt3}.$
$r=\dfrac{\cancel{2}\times\frac{\cancel{a}\times a\sqrt3}{\cancel{4}_2}}{3\cancel{a}}=\dfrac{a\sqrt3}{6}.$
$\therefore R^2-2Rr=\dfrac{a^2}{3}-\cancel{2}\times \dfrac{a}{\cancel{\sqrt3}}\times\dfrac{a\cancel{\sqrt3}}{\cancel{6}_3}=\dfrac{a^2}{3}-\dfrac{a^2}{3}=\boxed{0}.$