here&#x27;s a proof that the ratio of the triangle area to the square is scale invariant:<p>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.<p>the area of each of the original squares is s^2, and the new area is (ks)^2 = k^2*s^2<p>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.<p>(we could&#x27;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)