Prove that (2a,4a), (2a,6a) and (2a+a root3,5a) are vertices of an equilateral triangle.

Answer 1

just show that the sides have the same length.

For instance, the first two points define a side of length d, where

d^2 = (2a-2a)^2+(6a-4a)^2 = 4a^2
so, d=2a

Now just show that the other two sides are the same length.

Answer 2

We could "shrink" the figure, after all sides of all equilateral triangles are in the same ratio, so let's use
A(2,4), B(2,6), and C(2+√3 , 5)

are the angles equal to 60° each ??

clearly AB is a vertical line.
slope CA = 1/(2+√3 - 2) = 1/√3
so CA makes an angle of 60° with the x-axis
slope CB = -1/√3, so CB makes an angle of 120°

So , look at your sketch, what do your think ?

Btw, we could have found the slopes using the original points containing the "a" 's, they would have cancelled .