Centroids and some characterizations of parallelograms

Abstract

For a polygon P, we consider the centroid G0 of the vertices of P, the centroid G1 of the edges of P and the centroid G2 of the interior of P, respectively. When P is a triangle, the centroid G0 always coincides with the centroid G2. For the centroid G1 of a triangle, it was proved that the centroid G1 of a triangle coincides with the centroid G2 of the triangle if and only if the triangle is equilateral. In this paper, we study the relationships between the centroids G0,G1 and G2 of a quadrangle P. As a result, we show that parallelograms are the only quadrangles which satisfy either G0 = G1 or G0 = G2. Further- more, we establish a characterization theorem for convex quadrangles satisfying G1 = G2, and give some examples (convex or concave) which are not parallelograms but satisfy G1 = G2. ? 2016 Korean Mathematical Society.