Lippmann’s axiom and Lebesgue’s axiom are equivalent to the Lotschnittaxiom

Autor:	Victor Pambuccian, Celia Schacht
Rok vydání:	2019
Předmět:	Algebra and Number Theory Quadrilateral Parallel postulate Right angle Absolute geometry Algebraic geometry Lebesgue integration Combinatorics symbols.namesake symbols Vertex (curve) Geometry and Topology Axiom Mathematics
Zdroj:	Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry. 60:733-748
ISSN:	2191-0383 0138-4821
DOI:	10.1007/s13366-019-00445-y
Popis:	We prove that both Lippmann’s axiom of 1906, stating that for any circle there exists a triangle circumscribing it, and Lebesgue’s axiom of 1936, stating that for every quadrilateral there exists a triangle containing it, are equivalent, with respect to Hilbert’s plane absolute geometry, to Bachmann’s Lotschnittaxiom, which states that perpendiculars raised on the two legs of a right angle meet. We also show that, in the presence of the Circle Axiom, the statement “There is an angle such that the perpendiculars raised on its legs at equal distances from the vertex meet” is equivalent to the negation of Hilbert’s hyperbolic parallel postulate.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::d134bb0bee4189f3ab42cb71c6e63ddf https://doi.org/10.1007/s13366-019-00445-y Zobrazit plný text záznamu Full text from SpringerLink