Inmathematics, specifically inincidence geometry and especially inprojective geometry, acomplete quadrangle is a system of geometric objects consisting of any fourpoints in aplane, no three of which areon a common line, and of the sixlines connecting the six pairs of points.Dually, acompletequadrilateral is a system of four lines, no three of which pass through the same point, and the six points ofintersection of these lines. The complete quadrangle was called atetrastigm byLachlan (1893), and the complete quadrilateral was called atetragram; those terms are occasionally still used. The complete quadrilateral has also been called aPasch configuration, especially in the context ofSteiner triple systems.[1]
The six lines of a complete quadrangle meet in pairs to form three additional points called thediagonal 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; theline segments connecting these pairs are calleddiagonals. 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 theFano plane, afinite geometry in which the diagonal points of a complete quadrangle arecollinear, some authors have augmented the axioms of projective geometry withFano's axiom that the diagonal points arenot collinear,[2] while others have been less restrictive.
A set of contracted expressions for the parts of a complete quadrangle were introduced byG. B. Halsted: He calls the vertices of the quadrangledots, and the diagonal points he callscodots. The lines of the projective space are calledstraights, and in the quadrangle they are calledconnectors. The "diagonal lines" of Coxeter are calledopposite connectors by Halsted. Opposite connectors cross at a codot. The configuration of the complete quadrangle is atetrastim.[3]
As systems of points and lines in which all points belong to the same number of lines and all lines contain the same number of points, the complete quadrangle and the complete quadrilateral both formprojective configurations; in the notation of projective configurations, the complete quadrangle is written as (4362) and the complete quadrilateral is written (6243), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration.Theprojective dual of a complete quadrangle is a complete quadrilateral, and vice versa. For any two complete quadrangles, or any two complete quadrilaterals, there is a uniqueprojective transformation taking one of the two configurations into the other.[4]
Karl von Staudt reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a "harmonic property" could be based on concomitants of the quadrangle: When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line atprojective harmonic conjugate positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called aharmonic range. Through perspectivity and projectivity, the harmonic property is stable. Developments of modern geometry and algebra note the influence of von Staudt onMario Pieri andFelix Klein .
In theEuclidean plane, the four lines of a complete quadrilateral must not include any pairs of parallel lines, so that every pair of lines has a crossing point.
Wells (1991) describes several additional properties of complete quadrilaterals that involve metric properties of theEuclidean plane, rather than being purely projective. The midpoints of the diagonals arecollinear, and (as proved byIsaac Newton) also collinear with the center of aconic that istangent to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; theorthocenters of the four triangles formed in this way lie on a second line (known as the ortholine, Steiner line or Aubert line[5][6]), perpendicular to the one through the midpoints. Thecircumcircles of these same four triangles meet in apoint. In addition, the three circles having the diagonals as diameters belong to a commonpencil of circles[7] the axis of which is the line through the orthocenters.
Thepolar circles of the triangles of a complete quadrilateral form acoaxal system.[8]: p. 179