If P also lies on N, then this also inverts to an extended line.
This leads to a case with three lines, considered earlier.
Thus we now consider cases where L and M are parallel extended lines, and N is a circle.
An extended line "touches" L and M if and only if it is paralle to both. There are two such
lines touching N, but these may include L or M, which do not count as common tangents.
The circles touching both form a single family, with centres lying on the line equidistant
from each. They lie between L and M.
You can vary N by moving its centre Q, or the point P.
You can move the circle tangent to L and M by moving its centre C.