Comment by WCSTombs

> This proof assumes that the area a triangle is some function k c^2 of the hypotenuse c where k is constant for similar triangles.

Area in what units? "Square" units? But we're free to choose any unit we want, so I choose units where the triangle itself with hypotenuse H has area H^2 units. To justify that, I think the only thing we need is the fact that area scales as the square of length. (There's that word "square" again, which implies a specific shape that is actually completely arbitrary when talking about area. Perhaps it's better to say that "area scales as length times length.")

> To me that truth isn’t necessarily any less fundamental than the Pythagorean theorem itself.

I think the Pythagorean Theorem is surprisingly non-fundamental, in that you can get surprisingly far without it. It's surprising because we usually learn about it so early.