Show that for an isometry $T:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$, if
$$ X \mapsto \mathrm{dist}(X,T(X)) \quad (X \in {\mathbb{R}}^n)$$
is a constant map, $T$ is a parallel translation.
Show that for an isometry $T:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$, if
$$ X \mapsto \mathrm{dist}(X,T(X)) \quad (X \in {\mathbb{R}}^n)$$
is a constant map, $T$ is a parallel translation.