nt.number theory - Proof of "if a^2 + b^2 = c^2 then a*b*c is divisible by 60"

There is a very friendly discussion of how to derive your parameterization geometrically in the first chapter of Rational Points on Elliptic Curves by Silverman and Tate. The method they use is the one discussed in the wikipedia article that Rob H. linked to, but I feel that the exposition in Silverman and Tate's book is considerably better.

The basic idea is as follows. Suppose that you are given integers $(x,y,z)$, not all zero, such that $x^2+y^2=z^2$. Then dividing by $z^2$ gives you rational numbers $(frac{x}{z},frac{y}{z})$ which solve the equation $x^2+y^2=1$. So we have a rational point on the unit circle. Conversely, if we are given a rational point on the unit circle, then we can clear denominators to get an integral solution to $x^2+y^2=z^2$. So the problem of parameterizing the integer solutions to your equation is equivalent to the parameterization of the rational points on the unit circle. Silverman and Tate spend the first chapter of their book explaining, in a very down to earth and readable way, how one goes about parameterizing the rational points on the circle. In fact, they do much more. They show how to parameterize the rational points on any conic having rational points (that is, to any equation $ax^2+bxy+cy^2+dx+ey+f$ having at least one rational solution $(x_0,y_0)$).