Complex numbers, plane and geometry

[90nizam]

Homework Statement

There are three complex numbers a, b and c. Show that these propositions are equals.
1. ABC (triangle from the three points in complex plane) is equilateral (T₁).
2. j or j² is the solution for az² + bz + c = 0.
3. a² + b² + c² = ab + bc + ca

Homework Equations

There is a hint. Equilateral triangles made from the bases AB, BC, and CA have centres of gravity from which we can construct another equilateral triangle (T₂).

The Attempt at a Solution

T₁ and T₂ are equal triangles. They have the same heights and sides. I've tried to use the equation to solve quadratic equation (quadratic formula) and assumed j is a solution, hence j² is compliment of j or $$\bar{j}$$. I found ac-3b=1. I have no idea how to use the equation. Is my assumption correct? Or my approach to the question is wrong?

 

[HallsofIvy]

Is this an engineering class? That is, is "j" the imaginary unit, j²= -1?

 

[90nizam]

This is a mathematic class and we use i as the imaginary unit. I don't think that the teacher mistyped it.