Proving distance preserving map is affine linear

Question

Given a map $T: \mathbb{E^n} \rightarrow \mathbb{E^n}$ that preserves distance. Prove that it is affine linear. How does one do this? What's the approach? Looking at colinear points?

Answer 1

Hint: Prove that for any $n+1$-tuple of points $(e_0,\ldots,e_n)$ in general position in $\Bbb{E}^n$, the map $$\Bbb{E}^n\ \longrightarrow\ \Bbb{R}^{n+1}:\ p\ \longmapsto\ (|p-e_0|,\ldots,|p-e_n|),$$ is injective.