Various centroids of polygons and some characterizations of rhombi
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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 $P$. 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.
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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