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Pappus axiom

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If $l$ and $l'$ are two distinct straight lines and $A,B,C$ and $A',B',C'$ are distinct points on $l$ and $l'$, respectively, and if none of these is the point of intersection of $l$ and $l'$, then the points of intersection of $AB'$ and $A'B$, $BC'$ and $B'C$, $AC'$ and $A'C$ are collinear.

Figure: p071140a

The truth of Pappus' axiom is equivalent to the commutativity of the skew-field of the corresponding projective geometry. The Desargues assumption is a consequence of Pappus' axiom (Hessenberg's theorem), and at the same time Pappus' axiom is a degenerate case of the Pascal theorem. The axiom was proposed by Pappus (3rd century).


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References

[a1] O. Veblen, J.W. Young, "Projective geometry" , 1 , Ginn (1910)
How to Cite This Entry:
Pappus axiom. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pappus_axiom&oldid=32799
This article was adapted from an original article by P.S. ModenovA.S. Parkhomenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article