Pythagorean Theorem II : Relationship to the Parallel Axiom

피타고라스의 정리 II : 평행공리와의 관계

  • Received : 2019.09.04
  • Accepted : 2019.10.30
  • Published : 2019.10.31


The proposition that the parallel axiom and the Pythagorean theorem are equivalent in the Hilbert geometry is true when the Archimedean axiom is assumed. In this article, we examine some specific plane geometries to see the existence of the non-archimidean Hilbert geometry in which the Pythagorean theorem holds but the parallel axiom does not. Furthermore we observe that the Pythagorean theorem is equivalent to the fact that the Hilbert geometry is actually a semi-Euclidean geometry.


Supported by : 한국연구재단


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