Simplest way to solve mathematical equations in Python
You discount the best answer as unacceptable.
Your question is "I want a free Computer Algebra System that I can use in Python."
The answer is "SAGE does that."
Have you looked at maxima/macsyma? SAGE provides bindings for it, and that's one of the more powerful free ones.
Here is how to solve your original question using Python (via Sage). This basically clarifies the remark Paul McMillan makes above.
sage: a,b,c = var('a,b,c')sage: solve([a+b+c==1000, a^2+b^2==c^2], a,b,c)[[a == 1000*(r1 + sqrt(r1^2 + 2000*r1 - 1000000))/(r1 + sqrt(r1^2 + 2000*r1 - 1000000) + 1000), b == -1/2*r1 - 1/2*sqrt(r1^2 + 2000*r1 - 1000000) + 500, c == r1], [a == 1000*(r2 - sqrt(r2^2 + 2000*r2 - 1000000))/(r2 - sqrt(r2^2 + 2000*r2 - 1000000) + 1000), b == -1/2*r2 + 1/2*sqrt(r2^2 + 2000*r2 - 1000000) + 500, c == r2]]