this might be a very elemental question but it has been bothering me for a while. Must of the proofs I've seen of the Pythagorean Theorem involve showing that the areas of the squares with side length $a$ and $b$ add up to the area of the square with side length $c$. This is generally done by rearranging triangles. My problem with this type of proofs is that they only show that the areas must be the same but don't show that $a^2+b^2=c^2$.
Why must the area of a square with side $a$ be defined as $a^2$. Say for example that you had another way of measuring the surface of a square with a given side length (and it behaves as we would intuitively want area to behave). If this function is called $A$ then the visual proofs of the theorem would only show that $A(a)+A(b)=A(c)$.
So, does this type of proof works because we just happen to define area as we do, or does $A(a)+A(b)=A(c)$ must imply $a^2+b^2=c^2$?
Now, if $A(a)+A(b)=A(c)$ does imply $a^2+b^2=c^2$ that would mean that our function $A$ (which behaves as area does) must include the square of the side in its formula. For example $A(x)=kx^2, k>0$ (which does imply the pythagorean theorem). Are there other ways to define the surface of a square such that it behaves as it physically does? Would the visual proofs still be valid?