Rolf Struve

The Steiner–Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries

We prove that (a) a generalization of the Steiner–Lehmus theorem due to A. Henderson holds in Bachmann’s standard ordered metric planes, (b) that a variant of Steiner–Lehmus holds in all metric planes, and (c) that the fact that a triangle with two congruent medians is isosceles holds in Hjelmslev planes without double incidences of characteristic

Non-euclidean geometries: the Cayley-Klein approach

\ne 3

Zum Begriff Der Projektiv-Metrischen Ebene

≠3.

Endliche Cayley-Kleinsche Geometrien

In his 1854 Habilitationsvortrag Riemann presented a new concept of space endowed with a metric of great generality, which, through specification of the metric, gave rise to the spaces of constant curvature. In a different vein, yet with a similar aim, J. Hjelmslev, A. Schmidt, and F. Bachmann, introduced axiomatically a very general notion of plane geometry, which provides the foundation for the elementary versions of the geometries of spaces of constant curvature. We present a survey of these absolute geometric structures and their first-order axiomatizations, as well as of higher-dimensional variants thereof. In the 2-dimensional case, these structures were called metric planes by F. Bachmann, and they can be seen as the common substratum for the classical plane geometries: Euclidean, hyperbolic, and elliptic. They are endowed with a very general notion of orthogonality or reflection that can be specialized into that of the classical geometries by means of additional axioms. By looking at all the possible ways in which orthogonality can be introduced in terms of polarities, defined on (the intervals of a chain of subspaces of) projective spaces, one obtains a further generalization: the Cayley-Klein geometries. We present a survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries.