Fermat's two square theorem states that:
An odd prime p can be written as a sum of two squares if and only if p = 1 (mod 4)
Furthermore, such a solution is unique. Many proofs have been given of this fact, but most of them are very tricky.
A relatively simple proof of the existence part appeared in a survey of proofs of Fermat's theorem by Alexander Spivak, where the key point is presented in a visual manner, which I will present. The proof of uniqueness is much simpler, and I will present it, time permitting.