Going waaaay back a couple pages, here's a neat solution to level 20 of the Euclid game. I haven't seen it posted in the comments on the site.
First some geo. Consider where the outer tangent intersects the two circles, call them A and B. Draw the radii that connect points A and B to the centers. These radii are both perpendicular, so they're parallel. Now, imagine sliding segment AB down those radii, until one of the endpoints overlaps the center of the smaller circle. Let's call that CD, where C is the center of the smaller circle.
This segment CD is still perpendicular to the radii of the larger circle. Now, if the radii of the circle are r1 and r2 with r1 < r2, you can draw a circle of radius (r2 - r1) with the same center as the larger circle. Segment CD is then the tangent from that circle that passes through C.
So, you first make a segment of length r2-r1, then draw the circle described above. However, you still have to construct the specific tangent you want. So, here's the neat trick - if you draw the two possible tangents and the two radii that are perpendicular to those tangents, you get a cyclic quadrilateral, because the opposite angles sum up to 180 degrees. Find the midpoint of the segment connecting the centers, and draw the circle. Where it intersects the r2 - r1 circle is the tangent point. Then, you only have to construct the parallel line.
