Construct the internal and external tangent lines of any two disjoint circles.

The idea of these constructions is to enlarge and reduce the sizes of the circles so that the construction of a line tangent to a circle through a given point can be used.

Now, I find the lines tangent to the new (blue) circle through point C. That construction is as follows:

Find the midpoint of segment AC. Use this as the center of a circle.

Since every line tangent to circle A is perpendicular to any radius of circle A and since every inscribed triangle with the diameter as a side is a right triangle, the points of intersection of circle M and circle A are the points of tangency.

Now, we can go back to the original construction. The tangent lines that we have drawn will be translated up and down by the length of the radius of the smaller circle (since the blue circle was formed by shrinking the bigger circle by the the radius of the smaller).

To construct the internal tangent lines, extend the radius of the larger by the radius of smaller and repeat the constructions above.

The same constuctions will work for two congruent circles.