One night last week, as I lay in bed thinking about the challenges of the economy, terrorism, and global climate change, I thought to myself, What this world really needs is another proof of the Pythagorean theorem.
This pleasant diversion helped me to fall asleep for a few nights in a row, at least until I came up with the following approach. I call it a physicist's proof not only because it's less than mathematically rigorous, but also because the idea is to begin with a triangle of zero size and derive its law of growth. This is a more "dynamical" way of understanding the theorem than the traditional dissection methods.
Without further ado, here is the argument:
We label the sides by x and y, so the hypotenuse is some function f(x,y). We assume f(0,0) = 0. Now let y increase by an infinitesimal amount dy; then the small triangle is similar to the large right triangle, so that df/dy = y/f. Separating, we find f^2 = y^2 + C(x). Repeating the argument with the horizontal leg, we find f^2 = x^2 + D(y). Thus C(x) - x^2 = D(y) - y^2 = E, a constant, whence f^2 = y^2 + x^2 + E. But f(0,0) = 0 implies E=0, so f^2 = y^2 + x^2. QED.
I believe this can be made rigorous using (sin q)' = cos q and (cos q)' = -sin q (relations which can be proved without recourse to the Pythagorean theorem!)
Anyway, having found a solution, I poked around the web to compare it to the extant proofs. As I expected, my approach was not new. The Wikipedia entry on the Pythagorean theorem features this same basic argument, as does this page's amazing list. The idea is attributed to Michael Hardy of the University of Toledo, from 1988. Let this post serve to celebrate the idea's 20th Anniversary.
As long as I'm being unoriginal, let me also link to a cool math page from the website of Contra Costa Community College. Yesterday I was on this site by chance and happened to see two of my recent triangle puzzles!