In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio 2:4:5      (b)  1:2:3          (c) 1:2:4      (d)  2:4:3

Explanation

Let `a` is the side of the equilateral triangle.

Then, `Delta = sqrt3/4a^2`

`s = (a+a+a)/2 = (3a)/2`

Now, Inradius `(r) = Delta/s = (sqrt3/4a^2)/((3a)/2) = a/(2sqrt3)`

Circumradius`(R) = (abc)/(4Delta) = (a^3)/(sqrt3/a^2) = a/sqrt3`

Exradii `(r_1) = Delta/(s-a) = (sqrt3/4a^2)/(a/2) = sqrt3/2a`

`:. r:R:r_1 = a/(2sqrt3):a/sqrt3:(sqrt3a)/2 = 1:2:3`

So, option `(b)` is the correct option.