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http://dx.doi.org/10.14477/jhm.2019.32.5.241
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Pythagorean Theorem II : Relationship to the Parallel Axiom |

Jo, Kyeonghee
(Division of Liberal Arts and Sciences, Mokpo National Maritime Univ.)
Yang, Seong-Deog (Dept. of Math., Korea Univ.) |

Publication Information

Abstract

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.

Keywords

Pythagorean theorem; Parallel Postulate; Semi-Euclidean; Archimedean Axiom;

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Times Cited By KSCI :
2 (Citation Analysis)

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