Since we are on HN and talking of the Pythagorean theorem, I must mention E. W. Dijkstra's proof. https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E... he proves the generalization sgn(α + β - γ) = sgn(a² + b² - c²) also by using similar triangles. Incredibly neat.
Whoa, that IS neat. Thanks for sharing! A key part of his reformulation is π = α + β + γ (the sum of the internal angles of a triangle equal 2 right angles). That statement is, funny enough, equivalent to the parallel postulate.
Going on a tangent now: lately I've been thinking that the "dead horse" of the Pythagorean theorem is actually trying to tell us that flat (as opposed to fractal!) dimensions come from composable self-similarity (squares).
Yeah if the parallel postulate is out then this obviously doesn't stand. Starting from the North Pole, walking down the prime meridian to the equator then turning right and walking along the equator finally turning right to the north again at the right point so you walk through New Orleans creates a triangle on a sphere with three right angles...
