How do you find the center of two concentric circles with just a straightedge?
How do you find the center of two concentric circles with just a straightedge?
The Wikipedia article on Steiner constructions mentions it, but doesn't explain it, and the source linked is a book I don't have. This has come up in a practical project.
Here's a solution for circles with different radius that doesn't require right angle measurement or parallel lines:
- Draw a tangent on the larger circle
- Draw two tangents on the smaller circle that intersect where the first tangent touches the larger circle
- Draw two tangents on the larger circle where the tangents from step 2 intersect the larger circle opposite to the first tangent
- Find the intersection of the tangents from the 3rd step. A line from this to where the first tangent touches the larger circle must go through the center of the two circles.
- Repeat the above with the first tangent intersecting on a different point on the larger circle. The intersection of the lines from the 4th step is the center of the circle.
Edit: visual aid
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.
I'm still curious about the no-tangents solution, but for my specific application I could probably physically rest a straightedge or flat plane on the circle somehow.
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.
Scribe a tangent to either circle. A line perpendicular to the tangent that passes through the intersection of the tangent must pass through the centre. Do it twice and where those two lines intersect is the centre point.
Not sure if that will work for your practical application - getting it truly perpendicular might be hard.
Exactly. I could do a straightedge construction of 90 degrees, but not until I have a center, so I'm back at square one.
Since this is a practical application (and many straightedge problems don't allow measuring), you could do the same but with two parallel chords.
Bisect both chords and the line formed by those points must also pass through the centre.
It's a primitive tech project, so parallel lines aren't already available, and might be hard to construct just by neusis with twigs. Once I have a center it would be no problem, but you know, chicken and egg. Bisecting may or not be doable, depending on details - a piece of cordage could be marked with mud and doubled over on itself.
I'm pretty sure this sort of thing is why the ancients were so nuts about straightedge construction in the first place.