Legendre-like theorems in a general absolute geometry.

Silvia Pianta, Mario Marchi, Helmut Karzel

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


In this paper the axiomatic basis will be a general absolute plane A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the set of points and the set of lines, α the order structure and ≡ the congruence, and where furthermore the word “general” means that no claim is made on any kind of continuity assumptions. Starting from the classification of general absolute geometries introduced in [5] by means of the notion of congruence, singular or hyperbolic or elliptic, we get now a complete characterization of the different possibilities which can occur in a general absolute plane studying the value of the angle δ defined in any Lambert–Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yelds, in particular, a Archimedes-free proof of a statement generalizing the classical “first Legendre theorem” for absolute planes.
Original languageEnglish
Pages (from-to)61-71
Number of pages11
JournalResults in Mathematics
Publication statusPublished - 2007


  • Legendre theorem
  • absolute geometry
  • classification of absolute geometry


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