> 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.
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.