Well for third circle H(x,y):= (x-a_3)^2 + (y-b_3)^2 - (r_3)^2 , all of them intersect at a common point then H(x,y)=0 and F(x,y) - G(x,y)=0 and F(x,y) - H(x,y) =0 ..etc
Extending this to spheres would be adding another dimension z then H(x,y,z) would be (x-a_3)^2 + (y-b_3)^2 + (z-c_3)^2 -(r_3)^2 .
But i don't understand even in the case of circles or spheres, whether they intersect if yes then how will i find the point of intersection.
One way would be solving for F(x,y)= G(x,y) then we get a equation of a line something like ux + vy + c=0 where u,v,c can be got. Now i can get the intersection of line and circle . I substitute the value of y into x and then solve for quadratic . The condition would be b^2 -4ac for the quadratic.
Is there any way to do this for n spheres. It is getting tedious. Please throw some light or techniques.