# Pappus's hexagon theorem

## Geometry theorem / From Wikipedia, the free encyclopedia

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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.[1] 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.[2]

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 .[3] 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.