Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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VARIOUS CENTROIDS OF POLYGONS AND SOME CHARACTERIZATIONS OF RHOMBI

  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Kim, Wonyong (Department of Mathematics Chonnam National University) ;
  • Lee, Kwang Seuk (Yeosu Munsoo Middle School) ;
  • Yoon, Dae Won (Department of Mathematics Education and RINS Gyeongsang National University)
  • Received : 2016.02.04
  • Published : 2017.01.31
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Abstract

For a polygon P, we consider the centroid $G_0$ of the vertices of P, the centroid $G_1$ of the edges of P and the centroid $G_2$ of the interior of P. When P is a triangle, (1) we always have $G_0=G_2$ and (2) P satisfies $G_1=G_2$ if and only if it is equilateral. For a quadrangle P, one of $G_0=G_2$ and $G_0=G_1$ implies that P is a parallelogram. In this paper, we investigate the relationships between centroids of quadrangles. As a result, we establish some characterizations for rhombi and show that among convex quadrangles whose two diagonals are perpendicular to each other, rhombi and kites are the only ones satisfying $G_1=G_2$. Furthermore, we completely classify such quadrangles.

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References

  1. Edmonds and L. Allan, The center conjecture for equifacetal simplices, Adv. Geom. 9 (2009), no. 4, 563-576. https://doi.org/10.1515/ADVGEOM.2009.027
  2. R. A. Johnson, Modem Geometry, Houghton-Mifflin Co., New York, 1929.
  3. Kaiser and J. Mark, The perimeter centroid of a convex polygon, Appl. Math. Lett. 6 (1993), no. 3, 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 (2009), no. 1-2, 81-91. https://doi.org/10.1007/s00022-010-0027-1
  5. D.-S. Kim, Ellipsoids and elliptic hyperboloids in the Euclidean space $E^{n+1}$, Linear Algebra Appl. 471 (2015), 28-45. https://doi.org/10.1016/j.laa.2014.12.014
  6. D.-S. Kim and D. S. Kim, Centroid of triangles associated with a curve, Bull. Korean Math. Soc. 52 (2015), no. 2, 571-579. https://doi.org/10.4134/BKMS.2015.52.2.571
  7. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120. https://doi.org/10.1016/j.laa.2012.02.013
  8. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl. 438 (2013), no. 3, 1356-1364. https://doi.org/10.1016/j.laa.2012.08.024
  9. D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc. 50 (2013), no. 6, 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103
  10. D.-S. Kim, Y. H. Kim, and S. Park, Center of gravity and a characterization of parabolas, Kyungpook Math. J. 55 (2015), 473-484. https://doi.org/10.5666/KMJ.2015.55.2.473
  11. 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., To appear.
  12. D.-S. Kim and B. Song, A characterization of elliptic hyperboloids, Honam Math. J. 35 (2013), no. 1, 37-49. https://doi.org/10.5831/HMJ.2013.35.1.37
  13. Krantz and G. Steven, A matter of gravity, Amer. Math. Monthly 110 (2003), no. 6, 465-481. https://doi.org/10.1080/00029890.2003.11919985
  14. Krantz, G. Steven, J. E. McCarthy, and H. R. Parks, Geometric characterizations of centroids of simplices, J. Math. Anal. Appl. 316 (2006), no. 1, 87-109. https://doi.org/10.1016/j.jmaa.2005.04.046
  15. S. Stein, Archimedes, What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.

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