Signed Quantities and Quadrilaterals: Geometry without Cases
Ordinary angles and unsigned areas discard orientation, and much of the case analysis in plane geometry is the price of that loss. Signed quantities (directed angles, oriented areas and directed distances) keep the signs and so let one argument serve for every configuration. We illustrate this with results that hold uniformly for convex, concave and crossed quadrilaterals: the diagonal area formula, an identity expressing one diagonal in terms of the sides and the other diagonal, a test for a pair of parallel sides, a theorem on equidiagonal quadrilaterals (such a quadrilateral has a pair of equal opposite sides if and only if it has a pair of parallel opposite sides), and a single ratio of directed distances that decides whether the vertices lie on an ellipse, only on hyperbolas, or on a circle.