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

Victor Pambuccian; Horst Struve; Rolf Struve

doi:10.1007/s13366-015-0278-y

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

Victor Pambuccian, Horst Struve, Rolf Struve

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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 ≠ 3.

Original language	English (US)
Pages (from-to)	483-497
Number of pages	15
Journal	Beitrage zur Algebra und Geometrie
Volume	57
Issue number	2
DOIs	https://doi.org/10.1007/s13366-015-0278-y
State	Published - Jun 1 2016

Keywords

Bachmann’s ordered metric planes
Equal medians
Hjelmslev groups
Steiner–Lehmus theorem

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1007/s13366-015-0278-y

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T1 - The Steiner–Lehmus theorem and “triangles with congruent medians are isosceles” hold in weak geometries

AU - Pambuccian, Victor

AU - Struve, Horst

AU - Struve, Rolf

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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 ≠ 3.

AB - 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 ≠ 3.

KW - Bachmann’s ordered metric planes

KW - Equal medians

KW - Hjelmslev groups

KW - Steiner–Lehmus theorem

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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