I am interested in the algebraic/geometric way of finding the pythagorean triplets such that
$$a^2 + b^2 = c^2$$
$$a + b + c = 1000$$
I do the obvious
$$a + b = 1000 - (a^2 + b^2)^{1/2}$$
$$a^2 + b^2 = 1000^2 -2(1000)a - 2(1000)b +2ab + a^2 +b^2$$
$$2a + 2b - \frac{2ab}{1000} = 1000$$
$$a + b -\frac{ab}{1000} = 500$$
I have no idea what to do at this point. I can't separate the variables, and any geometric solution is beyond my reach. The only other thing I can think of is writing
$$a + b = 500 + \frac{ab}{1000} = 1000 - c$$
But this is going backwards