Geometric Interpretation of complex numbesr

[lolittaFarhat]

z1,z2,z3 are distinct complex numbers, prove that they are the vertices of an equilateral triangle if and only if the following relation is satisfied:

z1^2+z2^2+z3^2=z1.z2+z2.z3+z3.z1

so i shall show that |z1-z2|=|z1-z3|=|z2-z3|but i do not know how to start.

 

[mfb]

I moved the thread to the homework section.
Did you try to square the second equation, or multiply it with suitable complex conjugates of the expression, to see what happens? You'll get products of two numbers, which looks closer to the first equation.

 

[suremarc]

I'd start by simplifying the problem geometrically. The idea is to move $z_1, z_2, z_3$ through a series of rotations and translations in order to simplify the equations. For example, you could reduce the problem to the case where $z_1$ is a real number.