In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that if A , B , C {\displaystyle A,B,C} is one set of collinear points, and a , b , c {\displaystyle a,b,c} is another set of collinear points, then the intersection points X , Y , Z {\displaystyle X,Y,Z} of line pairs A b {\displaystyle Ab} and a B , A c {\displaystyle aB,Ac} and a C , B c {\displaystyle aC,Bc} and b C {\displaystyle bC} are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon A b C a B c {\displaystyle AbCaBc} . 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 considers a pappian plane containing a hexagon as just described but with sides A b {\displaystyle Ab} and a B {\displaystyle aB} parallel and also sides B c {\displaystyle Bc} and b C {\displaystyle bC} parallel (so that the Pappus line u {\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 u {\displaystyle u} and the lines g , h {\displaystyle g,h} 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 A , B , C {\displaystyle A,B,C} , and another set of concurrent lines a , b , c {\displaystyle a,b,c} , then the lines x , y , z {\displaystyle x,y,z} defined by pairs of points resulting from pairs of intersections A ∩ b {\displaystyle A\cap b} and a ∩ B , A ∩ c {\displaystyle a\cap B,\;A\cap c} and a ∩ C , B ∩ c {\displaystyle a\cap C,\;B\cap c} and b ∩ C {\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 A B C {\displaystyle ABC} and a b c {\displaystyle abc} . This configuration is self dual. Since, in particular, the lines B c , b C , X Y {\displaystyle Bc,bC,XY} have the properties of the lines x , y , z {\displaystyle x,y,z} of the dual theorem, and collinearity of X , Y , Z {\displaystyle X,Y,Z} is equivalent to concurrence of B c , b C , X Y {\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.
Proof: affine form
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. Because of the parallelity in an affine plane one has to distinct two cases: g ∦ h {\displaystyle g\not \parallel h} and g ∥ h {\displaystyle g\parallel h} . The key for a simple proof is the possibility for introducing a "suitable" coordinate system: Case 1: The lines g , h {\displaystyle g,h} intersect at point S = g ∩ h {\displaystyle S=g\cap h} . In this case coordinates are introduced, such that S = ( 0 , 0 ) , A = ( 0 , 1 ) , c = ( 1 , 0 ) {\displaystyle \;S=(0,0),\;A=(0,1),\;c=(1,0)\;} (see diagram).
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