View source for Complete quadrangle

{{Short description|Geometric figure made of 4 points connected by 6 lines}}
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[[Image:Complete-quads.svg|thumb|300px|A complete quadrangle (at left) and a complete [[quadrilateral]] (at right).]]

In [[mathematics]], specifically in [[incidence geometry]] and especially in [[projective geometry]], a '''complete quadrangle''' is a system of geometric objects consisting of any four [[Point (geometry)|points]] in a [[Plane (geometry)|plane]], no three of which are [[Collinearity|on a common line]], and of the six [[Line (geometry)|lines]] connecting the six pairs of points. [[Duality (projective geometry)|Dually]], a ''complete [[quadrilateral]]'' is a system of four lines, no three of which pass through the same point, and the six points of [[Line–line intersection|intersection]] of these lines. The complete quadrangle was called a '''tetrastigm''' by {{harvtxt|Lachlan|1893}}, and the complete quadrilateral was called a '''tetragram'''; those terms are occasionally still used. The complete quadrilateral has also been called a '''Pasch configuration''', especially in the context of [[Steiner triple system]]s.<ref>{{cite journal
 | last1 = Grannell | first1 = M. J.
 | last2 = Griggs | first2 = T. S.
 | last3 = Whitehead | first3 = C. A.
 | doi = 10.1002/1520-6610(2000)8:4<300::AID-JCD7>3.3.CO;2-I
 | issue = 4
 | journal = Journal of Combinatorial Designs
 | mr = 1762019
 | pages = 300–309
 | title = The resolution of the anti-Pasch conjecture
 | volume = 8
 | year = 2000}}</ref>

== Diagonals ==
{{anchor|Fano's axiom}}
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 segment]]s 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,<ref>{{harvnb|Hartshorne|1967}}; {{harvnb|Coxeter|1987|p=15}}.</ref> 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'''.<ref>[[G. B. Halsted]] (1906) [https://archive.org/details/syntheticproject00halsuoft ''Synthetic Projective Geometry''], page 14 via [[Internet Archive]]</ref>

== Projective properties ==
[[Image:pappusharmonic.svg|thumb|right|200px|KLMN is a complete quadrangle;<br>''D'' is the [[projective harmonic conjugate]] of ''C'' with respect to ''A'' and ''B''.]]

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 configuration]]s; in the notation of projective configurations, the complete quadrangle is written as (4<sub>3</sub>6<sub>2</sub>) and the complete quadrilateral is written (6<sub>2</sub>4<sub>3</sub>), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration.
The [[Duality (projective geometry)|projective 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 unique [[projective transformation]] taking one of the two configurations into the other.<ref>{{harvnb|Coxeter|1987|p=51}}</ref>

[[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 at [[projective harmonic conjugate]] positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called a [[harmonic range]]. Through perspectivity and projectivity, the harmonic property is stable. Developments of modern geometry and algebra note the influence of von Staudt on [[Mario Pieri]] and [[Felix Klein]] .

== Euclidean properties ==
In the [[Euclidean 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.

{{harvtxt|Wells|1991}} describes several additional properties of complete quadrilaterals that involve metric properties of the [[Euclidean plane]], rather than being purely projective. The midpoints of the diagonals are collinear, and (as proved by [[Isaac Newton]]) also collinear with the center of a [[conic]] that is [[tangent]] to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; the [[orthocenter]]s of the four triangles formed in this way lie on a second line, perpendicular to the one through the midpoints. The [[circumcircle]]s 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]]<ref>Wells writes incorrectly that the three circles meet in a pair of points, but, as can be seen in [[Alexander Bogomolny]]'s animation of the same results, the pencil can be hyperbolic instead of elliptic, in which case the circles do not intersect.</ref> the axis of which is the line through the orthocenters.

The [[polar circle (geometry)|polar circles]] of the triangles of a complete quadrilateral form a [[coaxal circles|coaxal]] system.<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960).</ref>{{rp|p. 179}}

== See also ==
* [[Newton line]]
* [[Nine-point conic]]
* [[Quadrilateral]]
* [[Wirtinger sextic]]

== Notes ==
{{reflist}}

== References ==
* {{cite book
  | last = Coxeter | first = H. S. M.
  | author-link = H. S. M. Coxeter
  | title = Projective Geometry, 2nd ed.
  | year = 1987
  | publisher = Springer-Verlag
  | isbn = 0-387-96532-7
  }}
* {{cite book
  | last = Hartshorne | first = Robin
  | author-link = Robin Hartshorne
  | title = Foundations of Projective Geometry
  | year = 1967
  | publisher = W. A. Benjamin
  | pages = 53–6
  }}
* {{cite book|first=Robert|last=Lachlan|url=http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=Lach015|title=An Elementary Treatise on Modern Pure Geometry|location=London, New York|publisher=Macmillan and Co.|year=1893}} Link from [[Cornell University]] Historical Math Monographs. See in particular tetrastigm, page 85, and tetragram, page 90.
* {{cite book
  | last = Wells | first = David
  | title = The Penguin Dictionary of Curious and Interesting Geometry
  | url = https://archive.org/details/penguindictionar0000well | url-access = registration | publisher = Penguin
  | year = 1991
  | pages = [https://archive.org/details/penguindictionar0000well/page/35 35–36]
  | isbn = 0-14-011813-6
  }}

== External links ==
* {{springer|title=Quadrangle, complete|id=p/q076010}}
* {{cite web
  | last = Bogomolny | first = Alexander | author-link = Alexander Bogomolny
  | title = The Complete Quadrilateral
  | work = [[Cut-the-Knot]]
  | url = http://www.cut-the-knot.org/ctk/CompleteQuadrilateral.shtml}}
* {{mathworld|urlname=CompleteQuadrangle.html|title=Complete Quadrangle}}

{{Incidence structures}}

[[Category:Projective geometry]]
[[Category:Configurations (geometry)]]
[[Category:Types of quadrilaterals]]