Desargues' Theorem

Desargues' Theorem Refers to the configuration above. If you start with 6 points--colored cyan, yellow, and magenta--placed in an arrangement like the one shown, then the circled intersection points will lie on a line.

Classical Proof: Consider a "tripod" T in 3-space formed by 3 coincident lines and the planes they span. Take two auxilliary planes and intersect them with T. This intersection will be a pair of triangles. The intersection points of corresponding edges of these triangles will lie on the line of intersection of the auxilliary planes. Projecting this configuration to the plane gives Desargues' theorem. To see this in action, click repeatedly on the figure below.