Assume a, b, c are the sides of a triangle which is not right, but for which nonetheless a2 + b2 = c2.

Position the sides a and b as to make a right angle. Let d be the distance between their endpoints. By the Pythagorean theorem,
a2 + b2 = d2. Which implies

c2 = d2.

From where, c = ±d. However, both c and d are positive from the context. Therefore, c = d in the first place. Therefore, the original triangle was in fact right.