In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
given one set of collinear points and another set of collinear points then the intersection points of line pairs and and and are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon .
It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes.
If one restricts the projective plane such that the Pappus line is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.
If the Pappus line and the lines have a point in common, one gets the so-called little version of Pappus's theorem.
The dual of this incidence theorem states that given one set of concurrent lines , and another set of concurrent lines , then the lines defined by pairs of points resulting from pairs of intersections and and and are concurrent. (Concurrent means that the lines pass through one point.)
Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem.
The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of and . This configuration is self dual. Since, in particular, the lines have the properties of the lines of the dual theorem, and collinearity of is equivalent to concurrence of , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.
If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique.
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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 objects, that is, things being "in a line" or "in a row". In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line".
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, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem.
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