“Robert P Crease explains why Pythagoras’s theorem is not simply a way of computing hypotenuses, but an emblem of the discovery process itself.”
From Physics Web:
Pythagoras’s theorem is important for its content as well as for its proof. But the fact that lines of specific lengths (3, 4 and 5 units, say) create a right-angled triangle was empirically discovered in different lands long before Pythagoras. Another empirical discovery was the rule for calculating the length of the long side of a right triangle (c) knowing the lengths of the others (a and b), namely c2 = a2 + b2.
Indeed, a Babylonian tablet from about 1800 BC shows that this rule was known in ancient Iraq more than 1000 years before Pythagoras, who lived in the sixth century BC. Ancient Indian texts accompanying the Sutras, from between 100 and 500 BC but clearly passing on information of much earlier times, also show a knowledge of this rule. An early Chinese work suggests that scholars there used the calculation at about the same time as Pythagoras, if not before.
But what we do not find in these works are proofs – demonstrations of the general validity of a result based on first principles and without regard for practical application. Proof was itself a concept that had to be discovered. In Euclid’s Elements we find the first attempt to present a more or less complete body of knowledge explicitly via proofs.
More here.