Start with two given circles, such as the two below. (Note that this example is just one of several cases to consider. The constructions for each case are similar.)

Next, choose a random point on one of the circles, say the larger one for this case. Construct a line through the random point and the circle's center. This is the dashed line in the below pictures. The center of the common tangent circle will lie somewhere on this line.

Next, construct a circle congruent to the smaller circle. Place the center at the chosen random point and use the radius of the smaller circle to construct. Also, construct the intersection of the dashed line with the newly formed circle.

The next step is to construct the segment from one of the intersection points with the center of the smaller circle. Then, construct the midpoint of this segment.

Construct the perpendicular to the segment at the midpoint. Find the intersection of the perpendicular with the dashed line. This point will be the center of the common tangent circle!

We now have the center, and we just need the radius to find the common tangent circle. The radius will be the length from this new center to the designated random point. The common tangent circle to the two given circles is shown in red!