Martin's theorem for Euclidean n-space and a generalization to the perimeter case (Q1077715)

The Beckman-Quarles theorem states that any mapping of Euclidean n-space \({\mathbb{R}}^ n\) \((1<n<\infty)\) into itself which preserves unit distances is a Euclidean motion. The author proves the following two theorems for Euclidean \({\mathbb{R}}^ n(2<n<\infty):\) Any mapping \(f: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) which preserves triangles of unit area is a Euclidean motion. The same holds for any injection \(g: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) which preserves triangles of unit perimeter. Both theorems are consequences of the Beckman-Quarles theorem, as the author points out by ingenious elementary reasoning.