Pappus's hexagon theorem

In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that

• given one set of collinear points ${\displaystyle A,B,C,}$ and another set of collinear points ${\displaystyle a,b,c,}$ then the intersection points ${\displaystyle X,Y,Z}$ of line pairs ${\displaystyle Ab}$ and ${\displaystyle aB,Ac}$ and ${\displaystyle aC,Bc}$ and ${\displaystyle bC}$ are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon ${\displaystyle AbCaBc}$.

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 ${\displaystyle Ab}$ and ${\displaystyle aB}$ parallel and also sides ${\displaystyle Bc}$ and ${\displaystyle bC}$ parallel (so that the Pappus line ${\displaystyle u}$ is the line at infinity), one gets the affine version of Pappus's theorem shown in the second diagram.

If the Pappus line ${\displaystyle u}$ and the lines ${\displaystyle g,h}$ 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 ${\displaystyle A,B,C}$, and another set of concurrent lines ${\displaystyle a,b,c}$, then the lines ${\displaystyle x,y,z}$ defined by pairs of points resulting from pairs of intersections ${\displaystyle A\cap b}$ and ${\displaystyle a\cap B,\;A\cap c}$ and ${\displaystyle a\cap C,\;B\cap c}$ and ${\displaystyle b\cap C}$ 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 ${\displaystyle ABC}$ and ${\displaystyle abc}$.^[3] This configuration is self dual. Since, in particular, the lines ${\displaystyle Bc,bC,XY}$ have the properties of the lines ${\displaystyle x,y,z}$ of the dual theorem, and collinearity of ${\displaystyle X,Y,Z}$ is equivalent to concurrence of ${\displaystyle Bc,bC,XY}$, 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.