Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

VARIOUS CENTROIDS OF QUADRILATERALS

  • Lee, Seul (Department of Mathematics, Chonnam National University) ;
  • Kim, Dong-Soo (Department of Mathematics, Chonnam National University) ;
  • Park, Hyeon (Department of Mathematics, Chonnam National University)
  • Received : 2017.02.24
  • Accepted : 2017.05.08
  • Published : 2017.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

For a quadrilateral P, we consider the centroid $G_0$ of the vertices of P, the perimeter centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P, respectively. It is well known that P satisfies $G_0=G_1$ or $G_0=G_2$ if and only if it is a parallelogram. In this note, we investigate various quadrilaterals satisfying $G_1=G_2$. As a result, for example, we show that among circumscribed quadrilaterals kites are the only ones satisfying $G_1=G_2$. Furthermore, such kites are completely classified.

Keywords

References

  1. Allan L. Edmonds, The center conjecture for equifacetal simplices, Adv. Geom. 9(4) (2009), 563-576. https://doi.org/10.1515/ADVGEOM.2009.027
  2. R. A. Johnson, Modem Geometry, Houghton-Miin Co., New York, 1929.
  3. Mark J. Kaiser, The perimeter centroid of a convex polygon, Appl. Math. Lett. 6(3) (1993), 17-19. https://doi.org/10.1016/0893-9659(93)90025-I
  4. B. Khorshidi, A new method for finding the center of gravity of polygons, J. Geom. 96(1-2) (2009), 81-91. https://doi.org/10.1007/s00022-010-0027-1
  5. D.-S. Kim and D. S. Kim, Centroid of triangles associated with a curve, Bull. Korean Math. Soc. 52(2) (2015), 571-579. https://doi.org/10.4134/BKMS.2015.52.2.571
  6. D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc., 50(6) (2013), 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103
  7. D.-S. Kim, W. Kim, K. S. Lee and D. W. Yoon, Various centroids of polygons and some characterizations of rhombi, Commun. Korean Math. Soc., 32(1) (2017), 135-145. https://doi.org/10.4134/CKMS.c160023
  8. D.-S. Kim, Y. H. Kim and S. Park, Center of gravity and a characterization of parabolas, Kyungpook Math. J. 55(2) (2015), 473-484. https://doi.org/10.5666/KMJ.2015.55.2.473
  9. 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(3) (2016), 637-645. https://doi.org/10.4134/CKMS.c150165
  10. Steven G. Krantz, A matter of gravity, Amer. Math. Monthly 110(6) (2003), 465-481. https://doi.org/10.2307/3647903
  11. Steven G. Krantz, John E. McCarthy and Harold R. Parks, Geometric characterizations of centroids of simplices, J. Math. Anal. Appl. 316(1) (2006), 87-109. https://doi.org/10.1016/j.jmaa.2005.04.046
  12. S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.