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 possible to prove Desargues' theorem from a two dimensional perspective. Coexter and Whitehead also set up different systems of axioms for projective geometry which allow the result to be proved. Some authors
[Projective Geometry, Finite and Infinite, Brendan Hassett,
just take Desargues' theorem as an axiom, and add it to axioms 1,2,3 & 4 above, to define projective geometry.
We take a simpler approach in our proof, and imagine that our projective plane is embedded in a three dimensional projective space. This allows us to make a very straight forward argument for the validity of Desargues' theorem, similar to the one found in
[Projective Geometry: Creative Polarities in Space and Time, Olive Whicher].
The converse of our stated result (i.e., that when the points formed by linking corresponding edges of a triangle lie upon a straight line, then the triangles are perspective to one another), is also true. And some consider this to be part of Desargues' theorem. Although we only prove that perspective triangles imply colinear meeting points of corresponding edges.
Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.