Let be a projective plane. Two triangles Di=(ai,bi,ci) are central to a point z if , , and , , .

Dually, the two triangles are axial to a line a if for i=1,2, , , and , , , .

The plane is (a,z)-desarguesian if given any pair of triangles Di= (ai,bi,ci) (i=1,2) central to z with

the two triangles D1 and D2 are already axial with respect to a.

Finally, is desarguesian if is (a,z)-desarguesian for all points z and all lines a.

Theorem. Let be a projective plane, . Then is (a,z)-desarguesian if and only if is (a,z)-transitive.

Theorem. A projective plane is desarguesian if and only if there exists a three-dimensional vector space V over a skewfield K such that where PG(V) denotes the projective plane over V.


Contributed by Hauke Klein
Version $Id: desargues.html,v 1.1 2001/01/29 11:14:26 hauke Exp $