Projective Geometry 4 Desargues' Theorem Proof

Desargues' theorem is one of the most fundamental and beautiful results in projective geometry. Desargues' theorem states that if you have two triangles which are perspective to one another then the three points formed by the meets of the corresponding edges of the triangles will be colinear. We give an intuitive proof which is based at imagining this two dimensional situation from a three dimensional perspective. Desargues' theorem is true on the projective plane. The projective plane can be thought of as the `extended' euclidean plane - i.e., the familiar 2D space, with extra `ideal' points at infinity, where parallel lines meet. The video's argument is not rigorous because we have not yet explained the axioms behind projective geometry. Our three dimensional argument quickly provides good evidence for Desargues' theorem, but if one wishes to prove it purely from within the two dimensional projective plane, one has to do quite some ground work. Projective geometry can be set up using the following axioms 1: Two distinct points lie on a unique line. 2: Two lines meet at a unique point. 3: There exist three non-colinear points. 4: Every line contains at least three points. (co-inear points are points that are on the same line.) However the four axioms above are not actually enough to establish Desargues' theorem, working purely inside the projective plane. Adding extra axioms, such as the projectivity axiom [Introduction To Projective Geometry, C.R. Wylie] make it...