Pythagorean triples

Zoom in/out and move the red color upper-right vertex of the envelope rectangle to change the size and shape of the visible area and find green dots representing the legs of Pythagorean triangles.
Why do the green dots representing the legs of Pythagorean triangles lie on the shown green parabolas? Why do these dots lie on the shown blue hyperbolas? The parametric set of green parabolas satisfies the equations: y² = 2nx + n² or x² = 2ny + n², where n represents a natural parametric variable. The parametric set of blue hyperbolas satisfies the equations: xy + 2n² = 2n (x + y), where n again represents a natural parametric variable.