Equilateral Triangles and Complex Variables: Proving the Relationship

[nicksauce]

Homework Statement

Let {z1,z2,z3} be complex variables such that |z1| = |z2| = |z3|. Prove that z1,z2,z3 are vertices of an equilateral triangle iff z1 + z2 + z3 = 0.

Homework Equations

The Attempt at a Solution

Not really sure where to start on this. I know that |z2-z1= |z3-z2| = |z3-z1|, but this information didn't get me very far. Any hints on how I should start this proof, or what other information I will need?

 

[Dick]

Divide your equation by z1. Now you have 1+z2/z1+z3/z1=0. So you can just work with the case 1+y1+y2=0 and |y1|=1 and |y2|=1. Does that make it seem easier?

 

[nicksauce]

Thanks that was quite helpful. Just one thing though... in the proof I needed to say that there exists just one equilateral triangle with (1,0) as a vertex, and has all the sides length 1 away from the origin. Is this as obvious as it intuitively seems to me, or do you think I should try to prove it?

 

[Dick]

Thanks that was quite helpful. Just one thing though... in the proof I needed to say that there exists just one equilateral triangle with (1,0) as a vertex, and has all the sides length 1 away from the origin. Is this as obvious as it intuitively seems to me, or do you think I should try to prove it?

It may seem obvious, but you still have to prove it. If you have 1+y1+y2=0 and |y1|=|y2|=1, look the the real and imaginary parts of y1 and y2. You can actually solve for them.