Hjelmslev's theorem

The set of Hjelmslev (also Hjelmslevscher centerline set called, in the English literature as Hjelmslev 's theorem known ) is a set of the geometry of the plane, which the Danish mathematician Johannes Hjelmslev - back (1873, 1950 ). Hjelmslev formulated this theorem in the context of his famous treatise on a new justification of the planar geometry, in which he shows that a planar geometry using only planar axioms, without continuity considerations, regardless of the parallel question can be established. In § 2 of the Treatise ( congruence and symmetry) while employed investigations into the flat Kongruenzabbildungen culminate in the set of Hjelmslev, which deals with a fundamental property of this Kongruenzabbildungen.

Wording of the sentence

Given a congruence, as well as two straight and in the Euclidean plane.

For each point and its image point of the midpoint of the segment is.

Then:

Either

Or