# Complex numbers, plane and geometry

1. Feb 16, 2010

### 90nizam

1. The problem statement, all variables and given/known data
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

2. Relevant 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).

3. 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?

Last edited: Feb 16, 2010
2. Feb 16, 2010

### HallsofIvy

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

3. Feb 16, 2010

### 90nizam

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