Desargues' Theorem

Deasargues' Theorem asserts: If we are given generic triples of points A, B, C and A', B', C' in the plane, then the intersections AB.A'B', AC.A'C', and BC.B'C' all lie on one line. The simplest way to see this result is to visualize the figure as the projection of one (say with the same labels) in 3D. In this case the assertion is clear, since the line is the intersection of the two 3D planes spanned by the points A, B, C and A', B', C'.

Desargues' configuration

In this figure A, B, C are the vertices of one triangle, and A', B', C' are the vertices of the other.