Rotations/Transformations with Complex Numbers/Eulers Formula

Question

Hello,

I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral triangle theta is $\pi/6$ but I don't know where to go from this?

Any help will be appreciated, thanks.

Answer 1

Hint:

I f a vertex of the triangle is represented by the complex number $a$, the other vertex are obtained from $a$ by rotations , about the origin, of angles $2\pi/3$ and $4\pi/3$. So they are: $b=a e^{2i\pi/3}$ and $c=a e^{4i\pi/3}$.

Use this in your formula and you can verify the result.

Answer 2

Hint: Assuming the triangle is centered at $0$, the radii from the origin to each of the corners are separated by angles $2\pi/3$. Pick one of the points to be $re^{it}$. The the other two points, in counterclockwise order, are $re^{it+2\pi i/3}$ and $re^{it+4\pi i/3}$.