How to prove that any bijective transformation of a Euclidean plane preserving distance $1$ is an isometry?
I can prove that it preserves natural distances: an equilateral triangle with side 1 is transformed into an equilateral triangle with side 1. So we can construct with three triangles the half of a regular hexagon, and it will be transformed into the half of a regular hexagon. So transformation preserves any natural distance.
But how to extend it to rational and real distances?