Commun. Korean Math. Soc. 2016; 31(3): 637-645
Printed July 31, 2016
https://doi.org/10.4134/CKMS.c150165
Copyright © The Korean Mathematical Society.
Dong-Soo Kim, Kwang Seuk Lee, Kyung Bum Lee, Yoon Il Lee, Seongjin Son, Jeong Ki Yang, and Dae Won Yoon
Chonnam National University, Yeosu Munsoo Middle School, Gwangju Munjeong Girls' High School, Gwangju Management High School, Chonnam National University, Gwangju Jeil High 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 $P$, respectively. When $P$ is a triangle, the centroid $G_0$ always coincides with the centroid $G_2$. For the centroid $G_1$ of a triangle, it was proved that the centroid $G_1$ of a triangle coincides with the centroid $G_2$ of the triangle if and only if the triangle is equilateral. In this paper, we study the relationships between the centroids $G_0, G_1$ and $G_2$ of a quadrangle $P$. As a result, we show that parallelograms are the only quadrangles which satisfy either $G_0= G_1$ or $G_0= G_2$. Furthermore, we establish a characterization theorem for convex quadrangles satisfying $G_1= G_2$, and give some examples (convex or concave) which are not parallelograms but satisfy $G_1= G_2$.
Keywords: center of gravity, centroid, polygon, triangle, quadrangle, parallelogram
MSC numbers: 52A10
2017; 32(1): 135-145
2018; 33(1): 237-245
2017; 32(3): 709-714
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd