VARIOUS CENTROIDS OF POLYGONS AND SOME CHARACTERIZATIONS OF RHOMBI

Abstract

For a polygon P, we consider the centroid Go 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 Go = G(2) and (2) P satisfies Gi = G(2) if and only if it is equilateral. For a quadrangle P, one of Go = G(2) and Go = G1 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.