On mappings preserving equilateral triangles (Q1882458)

The authors show that if a non-constant self-map \(\varphi\) of a finite-dimensional Euclidean space of dimension \(\geq 2\) preserves isosceles triangles (or if it is measurable and preserves equilateral triangles), then it is a similarity. If the dimension is \(\geq 3\) and \(\varphi\) is not constant and preserves equilateral triangles, then it is a similarity (the hypothesis of measurability being no longer needed). Expressed in terms of definability and axiomatizability inside first-order logic, the logical counterparts of these results were proved by \textit{M. Pieri} (1908), and \textit{E. W. Beth} and \textit{A. Tarski} [Nederl.\ Akad.\ Wet., Proc.\ Ser.\ A 59, 462--467 (1956; Zbl 0072.15503)] (see also \textit{W. Schwabhäuser, W. Szmielew} and \textit{A. Tarski} [Metamathematische Methoden in der Geometrie (Springer-Verlag, Berlin) (1983; Zbl 0564.51001)]).