complete quadrilateral

Complete quadrangle

In mathematics, specifically projective geometry, a complete quadrangle is a projective configuration (4₃6₂) consisting of four points, no three of which are collinear, and the six lines defined by those four points. This configuration was called a tetrastigm by Lachlan (1893), and that term is occasionally still used. The projective dual configuration (6₂4₃) to a complete quadrangle is a complete quadrilateral (called a tetragram by Lachlan), a configuration consisting of four lines, no three of which pass through a common point, and the six points of intersection of those four lines.

The six lines of a complete quadrangle meet in pairs to form three additional points beyond the four defining the configuration; it is one of the fundamental axioms of projective geometry that these three points can never be collinear (Coxeter 1987: 15). For any two complete quadrangles, or any two complete quadrilaterals, there is a unique projective transformation taking one of the two configurations into the other (Coxeter 1987: 51).

Wells (1991) describes several additional properties of complete quadrilaterals that involve metric properties of the Euclidean plane, rather than being purely projective. The six points of a complete quadrilateral form three pairs of points that are not already connected by lines; the line segments connecting these pairs are called diagonals. The midpoints of the diagonals are collinear, and (as proved by Isaac Newton) also collinear with the center of a conic tangent to the lines of the quadrilateral. Any three of the lines form a triangle; the orthocenters of these triangles lie on a second line, perpendicular to the one through the midpoints. The circumcenters of these same four triangles meet in a point. In addition, the three circles having the diagonals as diameters belong to a common pencil of circles the axis of which is the line through the orthocenters.