Assume we have the fully expanded circle equations
$C_1:x^2-2xc_x+c_x^2+y^2-2yc_y+c_y^2-R_1^2=0$ $C_2:x^2-2xh_x+h_x^2+y^2-2yh_y+h_y^2-R_2^2=0$
And we wish to solve this system for their intersection points.
There are 4 possible cases, no solution, exactly 2 solutions, exacly 1 solution, infinite solutions.
The biggest constraint however is that we need to solve this in a computer, thus symbolic manipulation is not allowed.
Is this possible?
Making $C_1-C_2$ we have the line which passes by both centers or
$$ L\to2(h_x-c_x)x+2(h_y-c_y)+c_x^2-h_x^2+c_y^2-h_y^2-R_2^2-R_1^2=0 $$
Now if $L$ is defined, $L \cap C_1$ and $L\cap C_2$. After that, analyzing the intersection points along $L$ should suffice to define the special case among the circles. If $L$ is undefined, then the circles are concentric.