This is kind of a fun one. The first time I ever learned about the formula for generating pythagorean triples, it was in the context of stereographic projections onto the unit circle with rational slopes. The visual intuition of that is nice, but the actual algebra to work out a formula is quite a mess, and (at least for me) the final result is easy to forget.
However, you can land on that same general formula just by squaring complex numbers with integer coordinates, which lends itself to a different visualization entirely. I still talk about the projections onto the unit circle, as it makes for a very nice little proof at the end, but I hope this offers a different perspective even to those familiar with the question of finding pythagorean triples.