In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio :

Let x be the length of each side of the equilateral triangle, then
$\begin{array}{l}\mathrm{\Delta }=\frac{\sqrt{3}{x}^2}{4},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}s=\frac{3x}{2}\\ r=\frac{\mathrm{\Delta }}{s}=\frac{x}{2\sqrt{3}},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}R=\frac{abc}{4\mathrm{\Delta }}=\frac{x}{\sqrt{3}}\end{array}$
$\begin{array}{l}{r}_1=r_2=r_3=\frac{\mathrm{\Delta }}{s-a}=\frac{x\sqrt{3}}{2}\\ \therefore r:R:r_1=1:2:3.\end{array}$