Quadrangle complet

In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. The six lines of a complete quadrangle meet in pairs to form three additional points called the diagonal points of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the line segments connecting these pairs are called diagonals. For points and lines in the Euclidean plane, the diagonal points cannot lie on a single line, and the diagonals cannot have a single point of triple crossing. Due to the discovery of the Fano plane, a finite geometry in which the diagonal points of a complete quadrangle are collinear, some authors have augmented the axioms of projective geometry with Fano's axiom that the diagonal points are not collinear, while others have been less restrictive. A set of contracted expressions for the parts of a complete quadrangle were introduced by G. B. Halsted: He calls the vertices of the quadrangle dots, and the diagonal points he calls codots. The lines of the projective space are called straights, and in the quadrangle they are called connectors. The "diagonal lines" of Coxeter are called opposite connectors by Halsted. Opposite connectors cross at a codot. The configuration of the complete quadrangle is a tetrastim.