here's a proof that the ratio of the triangle area to the square is scale invariant: imagine covering a geometric shape with a bunch of little squares, each wi

here's a proof that the ratio of the triangle area to the square is scale invariant: imagine covering a geometric shape with a bunch of little squares, each with side length s. you could imagine a limit argument, with the squares getting small enough to arbitrarily approximate the area of the shape. uniform scaling increases the distance between any two points by some factor k. the area of each of the original squares is s^2, and the new area is (ks)^2 = k^2*s^2 thus the area of the triangle increases by a factor of k^2. the same argument will apply to the square drawn on the hypotenuse. thus this k^2 term will cancel out when you take the ratio. (we could've used the area formula for a triangle here, but this argument applies more generally: for example we can deduce the area of a circle is proportional to radius^2 without deriving the formula)