Legendre-like theorems in a general absolute geometry
type: Item
instance of: scholarly article
title: Legendre-like theorems in a general absolute geometry
zbMATH Open document ID: 1142.51013
author: Helmut Karzel
author: Mario Marchi
author: Silvia Pianta
published in: Results in Mathematics
publication date: 2008-06-10T00:00:00Z
review text: Let \(A=(\mathcal{P},\mathcal{L},\alpha,\equiv)\) be an absolute plane, where \(\mathcal{P}\) is the set of points, \(\mathcal{L}\) is the set of lines, \(\alpha\) is the order structure, and \(\equiv\) is the congruence in \(A\). The authors show: Let \((a,b,c,d)\) be a Lambert-Saccheri quadrangle and \(\delta\) the angle \(\angle(c,d,a)\), then \(\delta=R\), \(\delta<R\), and \(\delta>R\) if and only if the congruence \(\equiv\) is singular, hyperbolic, and elliptic, respectively. Here \(2R\) denotes the straight angle. They obtain a similar theorem if \(\delta\) is replaced by \(\sigma\), where \(\sigma\) is the sum of angles in any triangle.
Mathematics Subject Classification ID: 51F05
Mathematics Subject Classification ID: 51F20
zbMATH DE Number: 5285854
zbMATH Keywords: absolute plane
zbMATH Keywords: Lambert-Saccheri quadrangle
zbMATH Keywords: sum of angles of a triangle.
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: s00025-007-0258-0
OpenAlex ID: W1975909754
DOI: 10.1007/S00025-007-0258-0
description: scientific article; zbMATH DE number 5285854