There is a very slick proof (discussed [here](https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-fermats-theorem) on MO) that every prime $p=4k+1$ is a sum of two squares, which looks at the set $S= \{(x,y,z) \in N^3: x^2+4yz=p \}$ and shows that a particular involution of $S$ has exactly one fixed point.