The Steiner-Lehmus theorem and "triangles with congruent medians are isosceles" hold in weak geometries

TL;DR

This paper extends classical triangle theorems, including Steiner-Lehmus and median-isosceles properties, to various non-Euclidean and weak geometric frameworks, demonstrating their broader applicability.

Contribution

It proves that key triangle properties hold in generalized and weak geometries such as Bachmann's ordered metric planes and Hjelmslev planes, broadening their theoretical scope.

Findings

Steiner-Lehmus theorem generalizes to Bachmann's ordered metric planes

A variant of Steiner-Lehmus holds in all metric planes

Triangles with congruent medians are isosceles in Hjelmslev planes without double incidences

Abstract

We prove that (i) a generalization of the Steiner-Lehmus theorem due to A. Henderson holds in Bachmann's standard ordered metric planes, (ii) that a variant of Steiner-Lehmus holds in all metric planes, and (iii) that the fact that a triangle with two congruent medians is isosceles holds in Hjelmslev planes without double incidences of characteristic $\neq 3$.