VARIOUS CENTROIDS OF QUADRILATERALS WITHOUT SYMMETRY

For a quadrilateral P, we consider the centroid G₀ of the vertices of P, the perimeter centroid G₁ of the edges of P and the centroid G₂ of the interior of P, respectively. It is well known that P satisfies G₀=G₁ or G₀=G₂ if and only if it is a parallelogram. In this paper, we investigate various quadrilaterals satisfying G₁G₂. As a result, we establish some characterization theorems. One of them asserts the existence of convex quadrilaterals satisfying G₁=G₂ without symmetry.