Unique point determined by 3 distances

[fk378]

Homework Statement

Suppose u,v,w are three complex numbers not all on the same line. Prove that any point in z in the complex plane is uniquely determined by its distances from these 3 points.

The Attempt at a Solution

I can explain this by saying something like, if u,v,w are points and given different radii for each of them, their circles will overlap and then one point is created. It's very messy, but I don't really know how to explain it. I also would like to show some sort of computation to prove it as well but can't seem to get my head around it.

 

[Mark44]

I haven't done this, but here is an approach, using a proof by contradiction. Instead of proving that any point in z in the complex plane is uniquely determined by its distances from u, v, and w, suppose that there are two points, z₁ and z₂ that are the same distances from u, v, and w. IOW, |z₁ - u| = |z₂ - u|, and so on for the distances to v and w.

If you arrive at a contradiction, that will prove what you really want to prove.