For a variation on this with fewer tangents (from A. S. Smogorzhevskii’s The Ruler In Geometrical Constructions):

• Pick point 1 on the larger circle.
• Draw two tangents (2) on the smaller circle, such that they go through point 1 and intersect the larger circle on the other end (3).
• Draw one line segment from 2 to 3’, and one from 3 to 2’. **
• Draw a line that goes through both the resulting intersection and the original point (1) you made on the larger circle. This line goes through the center of the circle.
• Repeat steps 1-4 from a different angle to get the center point.

The issue, of course, is that any tangent you draw (without other circles, lines, or tools) is going to be approximate, and so the center will also be approximate. Every solution for this that I found just assumes accurate tangents, or parallel lines, or whatever, but I don’t see a way to get those (I say, having only browsed through the topic briefly) when these two circles and a straightedge are all you have to work with. If that’s not a big deal in your practical application, cool.

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** I’m shortcutting, here. The long version is to first draw two line segments, one that uses the smaller circle’s tangent points (2) as endpoints, and one that uses the intersections on the larger circle (3) as endpoints. Because the two circles are concentric, these segments are parallel and centered on one another, so you end up with an isosceles trapezoid. You then draw its diagonals to get its midpoint.

Very cool, and thanks for the diagram!

That will work for me, I think, but drawing a tangent isn’t a standard straightedge operation. If Wikipedia is to be believed there’s still a “pure” solution to be found, just involving connecting intersection points.