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Pappus's hexagon theorem
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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
.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/Pappus-proj-ev.svg/220px-Pappus-proj-ev.svg.png)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Pappus-aff-ev.svg/220px-Pappus-aff-ev.svg.png)
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 considers a pappian plane containing a hexagon as just described but with sides and
parallel and also sides
and
parallel (so 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.