Global InformationLotschnittaxiom information
The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann.[1] It states:
Perpendiculars raised on each side of a right angle intersect.
Bachmann showed that, in the absence of the Archimedean axiom, it is strictly weaker than the rectangle axiom, which states that there is a rectangle, which in turn is strictly weaker than the Parallel Postulate, as shown by Max Dehn. [2] In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate.
- ^ Bachmann, Friedrich (1964), "Zur Parallelenfrage", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 27 (3–4): 173–192, doi:10.1007/BF02993215, S2CID 186240918.
- ^ Dehn, Max (1900), "Die Legendre'schen Sätze über die Winkelsumme im Dreieck", Mathematische Annalen, 53 (3): 404–439, doi:10.1007/BF01448980, S2CID 122651688