As we've stated it, Desargues's Theorem concerns triangles in the two dimensional extended plane P². However, there is also a more general version of Desargues's Theorem, which concerns triangles living in a three dimensional space. Surprisingly, this 3D version is easier to prove than the 2D version!
Now that we've defined the extended Euclidean space P³, let's state the 3D version of Desargues's Theorem.
Let's return to the shadow drawing challenge from the introduction video for Chapter Two. It will provide a key insight for proving Desargues's Theorem!
Upload your drawing to padlet by clicking the text box above! And then proceed to the next video, where we'll see one possible solution.
Let's see how the drawing challenge relates to Desargue's Theorem.
We've proved Desargues's Theorem for non-coplanar triangles in P³. But what about triangles in P³ that lie on a common plane?
We still need to prove the converse to Desargue's Theorem: if two triangles are in perspective from a line, then they must be in perspective from a point. We'll leave this proof as an exercise, but the following video provides some hints.