Projective geometry was invented by the French mathematician Desargues
(1591-1661) (for a biography in French, see
*http://bib1.ulb.ac.be/coursmath/bio/desargue.htm*). One of his theorems
is considered to be a cornerstone of the formalism. It states
that ``Two triangles are in perspective from a point if and only if they
are in perspective from a line'' (see fig. 2.1):

The theorem has a clear self duality: given two triplets of lines and defining two triangles, the intersections of the corresponding sides lie on a line if and only if the lines of intersection of the corresponding vertices intersect in a point.

We will give an algebraic proof: Let *P* be the common intersection of
*AA*', *BB*', *CC*'. Hence there are scalars
such that:

This indicates that the point on the line

the three intersection points are linearly dependent,