Legendre-like theorems in a general absolute geometry
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.
scientific article; zbMATH DE number 5285854