The formulas were known to Euclid and used by Diophantus to obtain Pythagorean triples with special properties. However, he never raised the question whether in this way one can obtain all possible triples. Conversely, all coprime triples can indeed be obtained in this manner. But the proof below only uses simple geometry and algebra. This is the well known equation of the unit circle with center at the origin.
Rational numbers approximate irrational to any degree of accuracy.
Therefore, the set of rational pairs is dense in the whole plane. So, perhaps, one might expect that any curve should contain a lot of rational pairs or meander wildly to avoid them. But this is not the case. But there are simpler examples. Moreover, pulling the unit circle even a little aside may change the picture drastically.