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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''inciden ...
and especially in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, a complete quadrangle is a system of geometric objects consisting of any four
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * 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 In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
'' is a system of four lines, no three of which pass through the same point, and the six points of
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
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.


Diagonals

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 In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
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 In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
, a finite geometry in which the diagonal points of a complete quadrangle are
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
, 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 George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
: 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.


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 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. The
projective dual In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of dua ...
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 In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
taking one of the two configurations into the other. 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 In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line ...
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 Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pie ...
and
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
.


Euclidean properties

In the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
, 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. describes several additional properties of complete quadrilaterals that involve metric properties of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
, rather than being purely projective. The midpoints of the diagonals are collinear, and (as proved by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
) also collinear with the center of a
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
that is
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; the
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
s of the four triangles formed in this way lie on a second line, perpendicular to the one through the midpoints. The
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
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 circlesWells writes incorrectly that the three circles meet in a pair of points, but, as can be seen in
Alexander Bogomolny Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...
's animation of the same results, the pencil can be hyperbolic instead of elliptic, in which case the circles do not intersect.
the axis of which is the line through the orthocenters. The polar circles of the triangles of a complete quadrilateral form a coaxal system.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960).


See also

* Newton line * Nine-point conic *
Quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...


Notes


References

* * * Link from
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
Historical Math Monographs. See in particular tetrastigm, page 85, and tetragram, page 90. *


External links

* * * {{Incidence structures Projective geometry Configurations (geometry) Types of quadrilaterals