Constructing the center of a circle, straightedge only. Variants on the Poncelet-Steiner Theorem

The Wikipedia Poncelet-Steiner Theorem entry says: Any Euclidean construction possible with straightedge and compass can be done with straightedge alone, provided that at least one circle with its center identified already exists on the plane. I already know this to be true as I've worked the constructions and seen the proof.

What interests me and what this question is about are some of the other claims made in the article - variants on the Poncelet-Steiner theorem. As it turns out the one circle with its center identified isn't strictly necessary. Alternative criteria exist, including:

1. Having two concentric circles without a center.

2. Having two distinct intersecting circles without their centers.

I have also worked these constructions and am satisfied. They are fairly straightforward. Proofs attributed to Detlef Cauer. They culminate in the construction of the center of the circle, thus reducing the scenario to the original Poncelet-Steiner hypothesis. No issue here so far.

But still other claims are made, between the Wikipedia, Wolfram's MathWorld, and other sources. Namely,

1. Two non-intersecting circles (without centers) but with an arbitrary point on the centerline (collinear with the centers).

2. Three non-intersecting circles.

3. I've also seen reference to two congruent circles with an arbitrary point on the meridian between them being sufficient. (thats a point on the perpendicular bisector of the segment connecting centers)

All three of these variants supposedly end in the construction of a circle's center point.

Another claim is that

1. Any arc of the circle with the center identified is enough (equivalent to the Poncelet-Steiner). Francesco Severi apparently gets credit.

But none of claims 3-6 have adequate citation. I've followed these citations to their inevitable dead-ends and I'm still left without proof or demonstration. For all of them.

I've tracked every source down, and the sources that they reference as well, right down to the bitter end, even accessing academic databases and checking out books at the library. I even bought one \$100 book that seemed promising. All they do is cite one another and vaguely refer to "his work", sometimes even quoting one another nearly verbatim. But never a proof. Never a construction.

One site I came across claims to provide a construction for one of these claims and to their credit they do show a lot, but it doesnt actually show the complete construction. In fact they use the phrases "you get the idea" and "a few hundred more lines to draw", leaving it to my imagination to fill in the gaps. Thats not even counting the fact the text is poorly worded and the images are improperly labeled.

And yes, I've tried proving them myself but unfortunately my intuition with geometry isn't that great. I also have no delusion about how complex or time-consuming these constructions might be.

Does anyone know how to prove variants 3-6? Or provide a citation to actual literature that explicitly proves or demonstrates these?

Because, as it stands, it would appear that these are claims made out of thin air with no real backing; assertions that have skimmed by without peer review. And, if so, that needs to be corrected. I have faith that these claims are true but I desperately want to learn them. The best of the best geometers I know cant tell me where to begin.

After a few years of searching I finally managed to put in the proper keyword combination to find something that lead me to something else. A book titled "Über die Konstruktion des Mittelpunktes eines Kreises mit dem Lineal allein", by Detlef Cauer, supposedly has the proof for three non-intersecting circles case (bullet point 4). Unfortunately the book is written in German and its hidden behind your standard academia (anti-education, anti-peer review, knowledge hording) paywall.

A translation of this texts proof(s) would be great, but that covers only one of the cases, leaving three still outstanding.

• Appreciate the constructive edits, @Blue. You write goodly. Mar 22, 2020 at 21:15
• What's the meridian in #5 ? Is it the line halfway between the 2 circles? Jul 12, 2020 at 17:00
• @brainjam Yes sir. Perpendicular bisector of the segment connecting centers. Jul 12, 2020 at 17:04
• I hesitate to even start an answer. You are asking for a detailed proof of 4 propositions. You refer to a site that explains #3 in some detail (I suspect enough to do the entire construction if you have the background), but are unsatisfied. You have a reference for #4 but imply that you want somebody to dig it up and translate if for you (translating math from German yourself is not insuperable even if you don't know the language). In my opinion, you are asking for an answer that would fill a chapter of a book. I respectfully suggest you narrow the scope of the answer you are requesting. Jul 12, 2020 at 17:19
• That said, I think it's a really interesting topic. I've learned a lot just looking around on the web for background. Jul 12, 2020 at 17:21

1 Answer

All of these constructions are fairly complex, and I'm not going to give them in any detail. OP has asked for information that will help move the ball forward. I'll give some comments and references, and will add detail or clarification or corrections where requested. I have not vetted any of the constructions in every detail. I'll number the claims to match the numbering in the question.

1. Two non-intersecting circles with a point on the join of the centers. The site referred to by OP is most likely http://mathafou.free.fr/pbg_en/sol100a.html. The construction is the topic of C. Gram. A remark on the construction of the centre of a circle by means of the ruler. Math. Scand., 4:157–160, 1956.

2. Three non-intersecting circles. Apparently in D. Cauer. Uber die Konstruktion des Mittelpunktes eines Kreises mit dem Lineal allein (Berichtigung). Math. Ann., 73(1):90–94, 1912 and 74(3):462–464, 1913 (German). If you can find Smogorzhevskii's The ruler in geometrical constructions, the construction for three circles not in the same coaxial system is given at the end of the book.

3. Two congruent non-intersecting circles with an arbitrary point on their radical axis. Use the following construction to get the line $$X_{11}X_{12}$$, which joins the circle center. After that use the same construction as in 3. In the figure, givens are in blue, and the points are numbered in order of creation. The four chords (e.g. $$X_1X_2$$) are polars of $$P$$ and X_6.

1. Any arc of the circle with the center identified. See Chris Impens' Geometry without compasses

2. Bonus case. Two non-congruent non-intersecting circles with an arbitrary point on their radical axis. As for case 5, build the points $$X_1,\dots,X_5$$. Let $$X_E$$ be the intersection of the lines $$X_2X_3$$ and $$X_1X_4$$. Then line $$X_5X_E$$ goes through the centers.

Akopyan and Fedorov's Two circles and only a straightedge gives a useful overview of some of these topics.

• Respect for going through with an answer after getting a vigourous response for misunderstanding what OP wanted when it wasn’t clearly stated in the question body Jul 13, 2020 at 5:14
• How did you even find these resources in such short notice, given that the topic is completely new to you? I am impressed. I havent vetted these for usefulness to my own ends... but this is far more than I previously had. If I solve any of the propositions I will eliminate them from my list and... if necessary... repost a new question eventually. Jul 14, 2020 at 3:48
• Google was helpful for 3 and 6. And 5 and 7 weren't difficult because 3 indicated you needed to find the center line, and because I studied some related constructions last year. I'm not sure why you think the topic is completely new to me. I was aware of Steiner constructions, but never paid much attention to them. If 4 relies on finding center lines there may be some hope of figuring that out without finding Cauer's paper... Jul 14, 2020 at 13:29
• @CalvinKhor Dont be a part of the problem. Dont deliberately antagonize or escalate when youre not even a part of the conversation. I made my question quite clear. Its not my fault this "community" look for any excuse at every turn to justify not giving an answer. I appreciate brainjam for actually answering the question, but at no point was clarification from me ever necessary. I should never have to justify asking a question. Jul 15, 2020 at 1:44
• I don't know what to tell you if you think that anyone was asking you to "justify asking a question". The math content of your question is crystal clear but I too struggled to understand what you wanted. Here are some great clarifications that came from your comments, which you could try to include in future questions: "Id be content with ONE detailed proof. One useful link."; "I [don't] expect one person to do all the work. Put in however much work you feel comfortable putting in. I just want to make progress in this endeavor."; "embedding a link, which is all I really asked for". Jul 15, 2020 at 2:30