Comment by Sesse__

> 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.

It is elementary to show that the area of a triangle is base * height / 2. (It follows from the fact that you can make a rectangle out of it using two identical sub-triangles. I assume you're willing to concede that the area of a rectangle is base * height.) If you scale your triangle by c, both base and height will be multiplied by c, and 2 will not.