Pappus's hexagon theorem

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

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

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

dual theorem: projective form

dual theorem: affine form

Because of the principle of duality for projective planes the dual theorem of Pappus is true:

If 6 lines $A,b,C,a,B,c$ are chosen alternately from two pencils with centers $G,H$ , the lines

are concurrent, that means: they have a point $U$ in common.
The left diagram shows the projective version, the right one an affine version, where the points $G,H$ are points at infinity. If point $U$ is on the line $GH$ than one gets the "dual little theorem" of Pappus' theorem.

If in the affine version of the dual "little theorem" point $U$ is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane.^[6] The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:

Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms connect, intersect and parallel, the statement is affinely invariant, and one can introduce coordinates such that $P=(0,0),\;Q=(1,0),\;R=(0,1)$ (see right diagram). The starting point of the sequence of chords is $(0,\lambda ).$ One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point.

If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.

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Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a , if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear.

permanent

In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:

(1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930

Coxeter, Harold Scott MacDonald

Cronheim, A. (1953), "A proof of Hessenberg's theorem", Proceedings of the American Mathematical Society, 4 (2): 219–221, :10.2307/2031794, JSTOR 2031794

doi

Dembowski, Peter (1968), Finite Geometries, Berlin: Springer-Verlag

Heath, Thomas (1981) [1921], A History of Greek Mathematics, New York: Dover Publications

Hessenberg, Gerhard (1905), "Beweis des Desarguesschen Satzes aus dem Pascalschen", Mathematische Annalen, 61 (2), Berlin / Heidelberg: Springer: 161–172, :10.1007/BF01457558, ISSN 1432-1807, S2CID 120456855

doi

Hultsch, Fridericus (1877), Pappi Alexandrini Collectionis Quae Supersunt, Berlin{{}}: CS1 maint: location missing publisher (link)

citation

Kline, Morris (1972), Mathematical Thought From Ancient to Modern Times, New York: Oxford University Press

Pambuccian, Victor; Schacht, Celia (2019), "The axiomatic destiny of the theorems of Pappus and Desargues", in Dani, S. G.; Papadopoulos, A. (eds.), Geometry in history, Springer, pp. 355–399, 978-3-030-13611-6

ISBN

Whicher, Olive (1971), Projective Geometry, Rudolph Steiner Press, 0-85440-245-4

Pappus's hexagon theorem

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permanent

Coxeter, Harold Scott MacDonald

doi

doi

citation

ISBN

ISBN

Pappus's hexagon theorem

Dual to Pappus's hexagon theorem

Pappus’s Theorem: Nine proofs and three variations