On mappings preserving equilateral triangles

Let E be a Euclidean space, dim E 2. We say that f : E E preserves equilateral triangles if for all triples of points x, y, z E we have We show that if E is a finite-dimensional Euclidean space, dim E 2, f:E E is measurable and preserves equilateral triangles, then it is a similarity transformation (an isometry multiplied by a positive constant). Moreover, in spaces at least three-dimensional we get a similarity transformation without any regularity assumption. Some generalizations as well as some interesting examples are also presented in the paper.