Commun. Korean Math. Soc. 2017; 32(1): 135-145
Online first article October 5, 2016 Printed January 31, 2017
https://doi.org/10.4134/CKMS.c160023
Copyright © The Korean Mathematical Society.
Dong-Soo Kim, Wonyong Kim, Kwang Seuk Lee, and Dae Won Yoon
Chonnam National University, Chonnam National University, Yeosu Munsoo Middle School, Gyeongsang National University
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.
Keywords: center of gravity, centroid, perimeter centroid, rhombus, kite, polygon, quadrangle
MSC numbers: 52A10
2016; 31(3): 637-645
2018; 33(1): 237-245
2017; 32(3): 709-714
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