There appear to be some tan,tan snaps for two circles, for the up to 4 tangent lines, when the
line starts "near" a tangent point....
There are 4 simultaneous equations, in the case of 2 circles...

http://www.cut-the-knot.org/Curriculum/Geometry/TangentTwoCirclesI.shtml#explanation


Even got tan,tan snap for a circle and a catenary... :-)