Characterizations of trapezoids (Q2871639)

This is another paper in a series of papers, written by the same author in the same journal, that deal with characterizations of special kinds of quadrilaterals. These include orthodiagonal, tangential, extangential, and bicentric quadrilaterals, and kites. In the paper under review, the author reviews eight known, and proves thirteen more, characterizations of trapezoids, i.e., figures in which two opposite sides are parallel. Some of these characterizations are trigonometric, some are metric, some refer to areas, and some refer to collinearities. For example, one of these characterizations states that if \(M\) is the point of intersection of the diagonals of a convex quadrilateral \(ABCD\), then \(AB\) is parallel to \(CD\) if and only if \(\sqrt{[ABCD]} = \sqrt{[ABM]} + \sqrt{[CBM]}\), where \([\,\cdot\,]\) denotes the area. Another characterization is given by the condition that the line \(\Lambda\) through the midpoints of \(AB\) and \(CD\) passes through \(M\).NEWLINENEWLINE This reviewer cannot but add the characterization given by the condition that the lines \(\Lambda\), \(BC\), and \(DA\) (when produced) are concurrent. This is equivalent to Archimedes' celebrated observation that if a triangle is dissected into line segments parallel to a side, then these midpoints are collinear -- an observation needed in proving that the center of mass \(M\) of a uniform triangular lamina lies on every median, and consequently that the three medians intersect, with their point of intersection being \(M\).