Software recommendation request: deciding whether a system of polynomial equations is solvable by radicals

The following system of equations comes from a very simple geometric figure I have to deal with a lot at work. Here $r_0,r_1,r_2$ and $L$ are known parameters, and the $x_i$s are the coordinates I'm after.

1. $x_0^2 + x_3^2 = r_0^2$
2. $(x_2-L)^2 + x_4^2 = r_2^2$
3. $x_3 = r_1 + \tfrac{1}{2}x_5(x_0 - x_1)^2$
4. $x_4 = r_1 + \tfrac{1}{2}x_5(x_2 - x_1)^2$
5. $x_0 + x_3 x_5(x_0 - x_1) = 0$
6. $(x_2-L) + x_4 x_5(x_2 - x_1) = 0$

I'm currently solving these numerically. This works perfectly well, but is (a) needlessly expensive if I don't have to do it and (b) extremely unsatisfying given the simple form of my equations.

My question is simply this: is there a piece of open-source (or otherwise free) software I can use to tell me - quickly and for certain - whether this system (and in future possibly others like it) can be solved explicitly in terms of radicals?