Look's like the general solution is called Monsky's Theorem: https://en.wikipedia.org/wiki/Monsky%27s_theorem. I would not have thought of that proof. Look's like that's the only known proof right now, too!
Posted January 27th, 2018
Yes, thanks, I knew it was out there, but thought it's an interesting result to share. I heard too that it has a generalization for higher dimensions, like cubes etc, which your citation mentions as well.
Here, I found Monsky's original paper/proof:
where he shows that in fact the areas of the dissected triangles satisfy some polynomial relation. His proof is just about 2 pages.
Posted January 28th, 2018
by The Fly