But we invented trigonometry (and algebra) so we didn't have to be clever about geometry any more! :)

I actually started out with a triangle dissection but I couldn't figure out the trick showing that you could pack exactly four triangles into the rectangle.

Now that I say that, it's probably something like using "all triangles with the same height and base have the same area" but I tend not to think in terms of non-rigid transformations.