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I am not a mathematician and I apologize if this question is naive.

I am interested in knowing how many distances are needed to determine the coordinates of a point in a Euclidean space of $n$ dimensions. I know no more than 3 distances are needed in a 2-dimensional space (trilateration). I am positive that no more than 4 distances are needed in a 3-dimensional space. I would think that $n+1$ distances are needed in an $n$-dimensional space, but I am not sure if this is the case, and most important, I do not know where to look for a proof.

Michele
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    Yes that is the case. It's simply and matter of solving the n+1 equations $(x_1 - a_{1,i})^2 + .... + (x_n - a_{n,i})^2 = d_i^2$. – fleablood May 20 '17 at 16:35
  • Many thanks. Any basic theorem/book I can reference to make this argument completely uncontroversial in a paper destined for a whole different audience? – Michele May 20 '17 at 16:38
  • Hmmm... maybe hilbert's geometry axioms. It's basically that two n-dimension hyperplanes meet and an n-1 dimensional plane and that to n-dimensional circles/spheres meet at an n-1 sphere. We need to get it down to an 0 dimensional circle (two points) and then an extra distance to determine which of the two points. I think a lot depends on the audience. – fleablood May 20 '17 at 16:58

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