Coincidences of centers of plane quadrilaterals (Q844009)

Associated with any (convex) planar quadrilateral one can define three natural centers of mass: the vertex centroid \(\mathcal G_0,\) the edge centroid \(\mathcal G_1,\) and the area centroid \(\mathcal G_2.\) For example, \(\mathcal G_1\) is defined as the center of mass of a system of four thin rods of uniform density and thickness placed along the sides of a quadrilateral. To this set of three centers of mass one can adjoin the Fermat-Torricelli center \(\mathcal F,\) defined as the point whose distances from the vertices have a minimum sum (in case of a general quadrilateral, this point is easily seen to be the intersection of diagonals). If, in addition, a quadrilateral is cyclic one can add the circumcenter \(\mathcal C\) to this list and if it is circumscriptible one can also consider the incenter \(\mathcal I.\) These are traditional centers of a convex quadrilateral and the authors investigate to what extent the pairwise equality of some of these centers determines the degree of regularity of the shape of the quadrilateral considered. It is well known that in a triangle the coincidence of any two traditional centers implies that the triangle is equilateral. The authors prove that if any two centers among \(\{\mathcal G_0, \mathcal G_1, \mathcal G_2, \mathcal F \},\) other than \(\mathcal G_1, \mathcal G_2\) coincide for a quadrilateral, then it is a parallelogram. Furthermore, in a cyclic quadrilateral \(\mathcal G_1 = \mathcal G_2\) if and only if it is a rectangle, or an isosceles trapezoid, each of whose equal legs is one third of the perimeter. For a circumscriptible quadrilateral \(ABCD\) denote by \(\alpha, \beta, \gamma, \delta\) respectively the lengths of tangent line segments drawn from the vertices \(A, B, C, D\) to the corresponding points of tangency with the incircle. Then for such quadrilateral that is not a rhombus the authors prove that any pairwise coincidence among \(\{\mathcal G_1, \mathcal G_2, \mathcal I \}\) implies coincidence of the other two pairs and is, in turn, equivalent to \(\delta = \beta = \alpha + \gamma\) or \(\alpha = \gamma = \beta + \delta.\) Further, \(\mathcal I = \mathcal G_0\) is equivalent to \(\delta + \beta = \alpha + \gamma\) and these four are the only coincidences among ten possible coincidences between pairs of its five centers \(\mathcal F, \mathcal G_0, \mathcal G_1, \mathcal G_2, \mathcal I\) that can occur.