Centroids and some characterizations of parallelograms
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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 $G_1= G_2$.
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 $P$, 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 $G_1= G_2$.
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