Common tangent to two circles with Ruler and Compass Given two circles (centers are given) -- one is not contained within the other, the two do not intersect -- how to construct a line that is tangent to both of them? There are four such lines.

[I will assume you know how to do basic constructions, and not explain (say) how to draw perpendicular from a point to a line.]
If you're not given the center of the circles, draw 2 chords and take their perpendicular bisector to find the centers $O_1, O_2$.
Draw the line connecting the centers. Through each center, draw the perpendicular to $O_1O_2$, giving you the diameters of the circles that are parallel to each other. Connect up the endpoints of these diameters, and find the point of intersection. There are 2 ways to connect them up, 1 gives you the exterior center of expansion (homothety), the other gives you the interior center of expansion (homothety).
Each tangent must pass through one of these centers of expansion. From any center of expansion, draw two tangents to any circle. Extend this line, and it will be tangential to the other circle.
