Not to bash the former president, but I'm failing to see what's so clever or nice about the proof... could someone please explain if I'm missing something? If you're going solve it with algebra on top of the similar triangles and geometry anyway, why complicate it so much? Why not just drop the height h and be done with it? You have 2 a b = 2 c h, c1/a = h/b, c2/b = h/a, c = c1 + c2, so just solve for h and c1 and c2 and simplify. So why would you go through the trouble of introducing an extra point outside the diagram, drawing an extra triangle, proving that you get a trapezoid, assuming you know the formula for the area of a trapezoid, then solving the resulting equations...? Is there any advantage at all to doing this? It seems to make strictly more assumptions and be strictly more complicated, and it doesn't seem to be any easier to see, or to convey any sort of new intuition... does it?