Editing Desargues's theorem (section)

==Self-duality==
By definition, two triangles are [[Perspective (geometry)|perspective]] if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be [[similarity (geometry)|similar]].

Under the standard [[duality (projective geometry)|duality of plane projective geometry]] (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration.<ref>{{harv|Coxeter|1964}}  pp.&nbsp;26–27.</ref>

This self-duality in the statement is due to the usual modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the [[theorem#Converse|converse]] of Desargues's theorem and was always referred to by that name.<ref>{{harv|Coxeter|1964|loc= pg. 19}}</ref>