It relates the lengths of the hypotenuse (side opposite the right angle) and legs (side adjacent to the right angle) of a right angle. In a right triangle, the sum of the squares. The lengths of the legs is equal to the square of the length of the hypotenuse.They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.
Derrick Boyd Geometry Blog 