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