So I recently stumbled upon a proof using stereographic projection to prove Euclid's formula for generating Pythagorean triples: for all $m,n\in \mathbb{N}$, $(2mn)^{2}+(m^{2}-n^{2})^{2}=(m^{2}+n^{2})^{2}$. Namely, for any rational point $P'=\frac{m}{n}\in \mathbb{Q}$, the stereographic projection of $P'$ onto the unit circle produces a point $P=(\frac{2mn}{m^{2}+n^{2}},\frac{m^{2}-n^{2}}{m^{2}+n^{2}})$. Then given $P$ is on the unit circle implies that $(\frac{2mn}{m^{2}+n^{2}})^{2}+(\frac{m^{2}-n^{2}}{m^{2}+n^{2}})^{2}=1$ and rearranging we get Euclid's formula. I have two questions:

1.) Does any one know any other cool and easy proofs like this one using stereographic projection?

2.) Is there some relationship between the points $P'$ and $P$ that make up a right triangle(say with $(0,0)$ or $(0,1)$ with sides $2mn$, $m^{2}-n^{2}$, and $m^{2}+n^{2}$? I assume that the geometry is there somewhere to make the proof work but I havent been able to figure it out myself.


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