In a short note in Greek, Nikolaos Kritikos proved in 1925 that the metric of a plane absolute geometry, in which there exists, for any two distinct points P and Q, an isometry \(\tau \) , referred to as a translation, satisfying \(A\tau (A)\equiv B\tau (B)\) for all points A and B, with \(\tau (P)=Q\) , must be Euclidean, i.e., there must exist a rectangle in that plane. The plane absolute geometry in this result is that of Hilbert’s Grundlagen der Geometrie, in which both order and free mobility are present. In this paper, we show that the result holds over Bachmann’s non-elliptic metric planes, even if just a single translation were to exist. This leads to three distinct versions of strengthenings of Kritikos’ result.
