Pythagorean Theorem I: In non-Hilbert Geometry

피타고라스의 정리 I: 비-힐베르트 기하에서

  • Received : 2018.08.23
  • Accepted : 2018.11.10
  • Published : 2018.12.31


Pythagorean thoerem exists in several equivalent forms in the Euclidean plane, that is, the Hilbert plane which in addition satisfies the parallel axiom. In this article, we investigate the truthness and mutual relationships of those propositions in various non-Hilbert planes which satisfy the parallel axiom and all the Hilbert axioms except the SAS axiom.


Supported by : 목포해양대학교, 한국연구재단


  1. H. EVES, An introduction to the History of Mathematics, Rinehart, New York, 1953.
  2. Robin HARTSHORNE, Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000.
  3. T. L. HEATH, The thirteen books of Euclid's Elements, translated from the text of Heiberg, with introduction and commentary, 2nd ed., 3 vols, University Press, Cambridge, 1926(Dover reprint 1956).
  4. D. HILBERT, The Foundations of Geometry, 2nd English Edition, Translated by Leo Unger from the 10th German Edition, Revised and Enlarged by Dr. Paul Bernays, The open court publishing company, 1971.
  5. K. JO, A historical study of de Zolt's axiom, Journal for History of Mathematics 30(5) (2017), 261-287.
  6. K. JO, S.-D. YANG, Moulton Geometry, Journal for History of Mathematics 29(3) (2016), 191-216.
  7. K. JO, S.-D. YANG, Pythagorean Theorem II : Relationship to Parallel Axiom, in preparation.
  8. R. KAYA, Area formula for Taxicab triangle, IIME Journal 12(4) (2006) 219-220.
  9. R. KAYA H. B. CALAKOGLU, Taxicab versions of some Euclidean theorems, Int. J. Pure Appl. Math. 26(1) (2006), 69-81.
  10. I. KOCAYUSUFOGLU, E. OZDAMAR, Isometries of Taxicab geometry, Commun. Fac. Sci. Univ. Ank. Series Al 47 (1998), 73-83.
  11. LEE Man-Guen, JEON Byung-Ki, comps., All that Pythagorean Proposition, Kyung Moon Sa, 2007. 이만근, 전병기 엮음, 올 댓 피타고라스의 정리, 경문사, 2007.
  12. E. S. LOOMIS, The Pythagorean Proposition, Classics in Mathematics Education Series., National Council of Teachers of Mathematics, 1968.
  13. George E. MARTIN, The foundations of geometry and the non-Euclidean plane, Springer, New York, 1975.
  14. F. R. MOULTON, A simple non-Desarguesian plane geometry, Transactions of the AMS April (1902), 192-195.
  15. M. OZCAN, R. KAYA, Area of a Taxicab triangle in terms of the Taxicab Distance, Missouri Journal of Mathematical Sciences 15(3) (Fall 2003), 21-27.
  16. K. P. THOMPSON, The nature of length, area, and volume in Taxicab geometry, arXiv:1101.2922v1.