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I am writing up some notes on equilateral triangles. I have reached the point where I want to show that a triangle is equilateral if and only if the three circles P, Q, and R, in the above diagram are congruent. I know the proof; my problem is I don't know how to refer to these circles. Is there a technical term I can use when referring to tangent circles such as these?

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Apollonian circles when included in bigger circles. I don't know about triangles. Maybe just inscribed circles. – ja72 Apr 12 '13 at 19:42
I would draw the picture, and just the label the circles by the letters you've just used. As they say, a picture can be worth a thousand words. Probably less, actually, but still better than trying to use some obscure term. – Christopher A. Wong Apr 12 '13 at 19:43
Related question:…. They mention kissing circles :-) – ja72 Apr 12 '13 at 19:49
@ChristopherA.Wong. What you suggest is probably what I will do if no one comes up with a succinct term. I was hoping there was some common term that I was ignorant of, not being a professional mathematician. – m_goldberg Apr 12 '13 at 19:50
@ja72. Thanks for the reference -- "kissing circles" may do although it's just a general synonym of "tangent circles" -- if nothing more specific comes up. – m_goldberg Apr 12 '13 at 19:55

I might as well try to get the accepted answer with "Kissing Circles", and "Apollonian Circles" and an alternate.

See related question Inscribed kissing circles in an equilateral triangle.


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Sorry and thanks for your contribution, but I like "2nd-order incircles" better. – m_goldberg Apr 13 '13 at 6:51
Great. Now you or someone needs to put this as an answer to the question. – ja72 Apr 13 '13 at 20:40
up vote 0 down vote accepted

Because it doesn't look like LordSoth is going to post an answer, I'm recording the suggestion he made for the record.

Of all terms suggested, I find second order in-circle the most descriptive of the concept I wanted to capture. I am using this term in the notes I am writing. The term has the advantage of generalization to n-th order in-circle, which would be a tangent circle added at the n-th stage of an Apollonian packing of a triangle. Indeed, the proposition stated in my question holds at any stage of an Apollonian packing.

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