DOI QR코드

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GENERALIZED VARIGNON'S AND MEDIAL TRIANGLE THEOREMS

  • 투고 : 2022.04.06
  • 심사 : 2022.07.28
  • 발행 : 2023.04.30

초록

In this paper, we extend the medial triangle theorem and Varignon's theorem to generic two-dimensional polygons and highlight the role played by diagonals in this process. One of the results is a synthetic definition of the concept of median for an n-sided polygon.

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참고문헌

  1. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, Inc., New York, 1961.
  2. H. S. M. Coxeter and S. L. Greitzer, Geometry revisited, New Mathematical Library, 19, Random House, Inc., New York, 1967.
  3. A. Emch, On the medians of a closed convex polygon, Amer. J. Math. 37 (1915), no. 1, 19-28. https://doi.org/10.2307/2370252
  4. Euclid, Euclid's Elements of Geometry, Edited and provided with a modern English translation by Richard Fitzpatrick, 2007.
  5. R. A. Johnson, Advanced Euclidean geometry: An elementary treatise on the geometry of the triangle and the circle, Dover Publications, Inc., New York, 1960.
  6. B. Khorshidi, A new method for finding the center of gravity of polygons, J. Geom. 96 (2009), no. 1-2, 81-91. https://doi.org/10.1007/s00022-010-0027-1
  7. D.-S. Kim, K. S. Lee, K. B. Lee, Y. I. Lee, S. Son, J. K. Yang, and D. W. Yoon, Centroids and some characterizations of parallelograms, Commun. Korean Math. Soc. 31 (2016), no. 3, 637-645. https://doi.org/10.4134/CKMS.c150165
  8. R. B. Kirchner, Classroom notes: A generalization of the median theorem for triangles, Amer. Math. Monthly 69 (1962), no. 7, 650. https://doi.org/10.2307/2310839
  9. N. Lord, 92.22 Maths bite: averaging polygons, Math. Gaz. 92 (2008), no. 523, 134.
  10. M. F. Mammana, B. Micale, and M. Pennisi, On the centroids of polygons and polyhedra, Forum Geom. 8 (2008), 121-130.
  11. E. E. Moise, Elementary Geometry from an Advanced Standpoint, third edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1990.
  12. P. N. Oliver, Consequences of the Varignon parallelogram theorem, The Mathematics Teacher 94 (2001), no. 5, 406-408.
  13. A. Ostermann and G. Wanner, Geometry by Its History, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-29163-0
  14. A. Palatnik, Proof without words: Varignon's theorem, College Math. J. 48 (2017), no. 5, 354. https://doi.org/10.4169/college.math.j.48.5.354
  15. G. C. Shephard, Centroids of polygons and polyhedra, Math. Gaz. 74 (1990), no. 467, 42-43. https://doi.org/10.2307/3618848
  16. J. Stillwell, The Four Pillars of Geometry, Springer, New York, 2005.
  17. P. Varignon, Elemens de Mathematique de Monsieur Varignon, Springer, Paris, 1731.