Plane quadrilaterals

The various types of plane quadrilaterals are characterized by their side and diagonal lengths. Pantographs are described. The set of all congruence classes of quadrilaterals is a variety of degree six in E6. There are in Euclidean plane geometry some elegant theorems about cyclic quadrilaterals, most of them consequences of the angle properties in circles. For example, there is Ptolemy’s theorem1: In a cyclic quadrilateral, the product of the diagonals equals the sum of the prod-ucts of opposite sides. For noncyclic quadrilaterals, results are not so easily come by. One general theorem of note is Euler’s2: In any quadrilateral, the sum of the squares on the four sides is equal to the sum of the squares on the diagonals plus four times the square on the line joining the midpoints of the diagonals. Euclid, in The Elements, devotes no more than passing interest in quadrilaterals which are not regular in some way, such as parallelograms. But later geometers have of course filled this gap, so that by the end of the nineteenth century the trigonometric properties of general quadrilaterals appear as suitable material for school textbooks, see [2], [1]. Nevertheless the emphasis was on cyclic quadrilaterals