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

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 $\ne 3$.