Double Balanced Quadrilaterals

The k-centroids Gk of a polygon, for k = 0, 1, 2, are the centroids of the polygon when the mass is equally distributed respectively between the vertices, along the perimeter, or across the area. A fundamental theorem by Al-Sharif, Hajja, and Krasopoulos in [1] asserts that the quadrilaterals with either G0 = G1 or G0 = G2 are precisely all parallelograms. Our main result describes the non-parallelograms with G1 = G2 by providing formulas for their diagonals in terms of the sides, as well as formulas for the ratios determined on the diagonals by their intersection point. In this way, we complete a fifteen-year-old problem by these three authors on characterizing all double balanced quadrilaterals. As an application, we show how our main theorem can be used to deduce their characterizations of double balanced circumscribed and cyclic quadrilaterals.