Construction of tangent.?

Let the green circles in this image be the given circles. They have centers A and B, and point of tangency T. Here the circles are shown externally tangent. There are three common tangent lines. The easiest one is the line through T perpendicular to line AB. The other two are a bit more complicated, but not much.

Construct radii AP and BQ, perpendicular to AB, with P and Q on the same side of AB.

Let PQ meet AB at point C. This is one of the points of homothety (point T being the other).

Construct point D, the midpoint of AC.

Construct a circle (the red one) centered on D, with diameter AC. Let it intersect circle A at points K and L.

Lines CK and CL are common tangents of the two given circles.

This same construction would hold up if the given circles were internally tangent. If the circles did not meet, a minor variation on this construction could produce all four common lines of tangency.

Followup:

I need to back away from one of those statements. If the circles were internally tangent, then they would have only the one common tangent through T.

Use your straightedge to draw in the radii to the point of tangency, then construct the perpendicular through this point.