Two more triangle centers

This note grew out of a question that a student asked while the first-named author was giving a lecture on triangle centers in a Geometry class in the fall of 2008. The student, Shefa’a Bani Melhem, wondered about the point in a given triangle that, when joined to the midpoints of the sides, divides the triangle into three quadrilaterals of equal area. Few days later, the aforementioned author asked her to prove that such a point is necessarily the centroid and to consider the point whose perpendiculars to the sides divide the triangle into three quadrilaterals of equal area. This problem turned out to be more difficult than was expected, and it is the purpose of this note to investigate the existence and uniqueness of such a point and to describe its trilinear coordinates (or simply, its trilinears), i.e., its