View source for Pappus axiom

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711402.png" /> are two distinct straight lines and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711404.png" /> are distinct points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711406.png" />, respectively, and if none of these is the point of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711408.png" />, then the points of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p0711409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p07114010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p07114011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p07114012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p07114013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071140/p07114014.png" /> are collinear.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071140a.gif" />

Figure: p071140a

The truth of Pappus' axiom is equivalent to the commutativity of the skew-field of the corresponding projective geometry. The [[Desargues assumption|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|Pascal theorem]]. The axiom was proposed by Pappus (3rd century).



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Veblen,   J.W. Young,   "Projective geometry" , '''1''' , Ginn  (1910)</TD></TR></table>