Complete quadrangle

Complete Quadrangle

Complete quadrangle

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In mathematics, specifically projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in the Euclidean plane, no three of which are on a common line, and of the six lines connecting each pair of points. 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. Since the discovery of the Fano plane, a non-Euclidean geometry in which the diagonal points of a complete quadrangle are collinear, it has become necessary to augment the axioms of projective geometry with Fano's axiom that the diagonal points are not collinear.

Projective properties

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 form projective configurations; in the notation of projective configurations, the complete...
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