Really a question for anyone interested in geometry, and it's really got me stumped. How do you find the equation (or even vector) of a line (either infinite or a segment) that is a common tangent to two circles?
There are many people willing to share methods for getting the lines length, but it's hard to get any other information. Good luck to any who try.
You mean in a situation where two circles are intersecting at exactly one point (I can't see any other way to have a single 'common' tangent line, but maybe I'm missing something obvious)?
In that case, draw the radius out from one of the circles (either one, as the tangent line will be the same) to where the circles intersect (calculate this point). Find the slope of that segment, and note that the tangent line is perpendicular to it. From there, you have the tangent line's slope and a line it passes through, so you can get its equation.