Equal Distances Between N Points in R^n-1+ | Solving System of Equations

[julypraise]

Is there a theorem that states that n distinct points in R^n-1 or higher one can be separated in an equal distance as the distance is greater than 0?

We know that 4 distinct points in R^2 cannot be positioned in an equal distance>0 but in R^3 it is possible as a pyramid shape.

If there is such a theorem, could you give me reference?

I've posted this on this area because it seems it is a problem of solving a system of equations.

Cheers

 

[Hawkeye18]

I do not know the reference, but it is a simple fact which is easy to prove:

Take in $\mathbb R^n$ n points (1, 0, ...,0), (0, 1, 0, ..., 0), ... (0, 0, ... , 0, 1) (the standard unit vectors). Then the distance between any 2 points is the same ($\sqrt 2$). Now take the (n-1) dimensional hyperplane through these points, and you get the points positioned in $\mathbb R^{n-1}$.

 

[julypraise]

Oh yeah... Thank you indeed!