I want to find the planes tangent to three given circles in 3D space.
I'm not sure how many solutions there are, in general. My guess is that there are 8. This suggests that we might have to find the roots of some polynomial of degree 8, which would be bad news.
A Google search for "plane tangent to three circles" yields exactly one result, which is this question. It was asked in 2011, and was not answered. Maybe the nasty notation scared people away, so let me suggest a nicer one:
Let's call the three circles $C_1$, $C_2$, $C_3$, and suppose that $C_i$ is defined by a center point $P_i$, a radius $r_i$, and a unit vector $N_i$ normal to its plane.
So, again, the question is:
find the equations of the tangent planes in terms of the $P_i$, $r_i$, and $N_i$.
The case where the three radii are equal is of some interest, if that's easier.
Also, I'm interested only in the case where the circles are in "general position", which means (I think) that the number of solutions is finite but non-zero. So, please feel free to ignore special cases like the circles having a common tangent line, or being coplanar, or lying on a common cylinder or cone, etc.