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Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture on the web, without any comment.

The chain of spheres is enveloped by a ring cyclide. The cyclide has (Yvon-)Villarceau circles, some of them are shown on the picture. We can see on the picture the following beautiful property: every Villarceau circle is tangent to each of the spheres of the chain. I have graphically checked that this property holds for any $n \geq 3$.

However, after googling during a couple of days, I've never found a statement of this property. Do you know a proof of this property, or a reference (book or article) providing this proof?

enter image description here

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    $\begingroup$ The original source of the image seems to be this page. The figure was apparently made with Cabri 3D by 鄭乃節 (Nai-Chieh CHENG), advised by 全任重 (Prof. Jen-Chung CHUAN) at NTHU in Hsinchu, Taiwan. $\endgroup$
    – j.c.
    Commented Jul 12, 2018 at 12:24

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22 October 2020

A link that outlines the essential mathematical machinery is here:

http://archive.bridgesmathart.org/2015/bridges2015-253.pdf

As stated in that abstract, a cyclide is the image of a torus under inversion in a sphere.

It is easy to see that if you place N congruent kissing spheres with centers along a suitable circular path, they implicitly define an enveloping torus tangent to each sphere along a great circle orthogonal to the plane of the torus.

Thus every Villarceau circle intersects each of these great circles at exactly one point, and that point on the Villarceau circle is tangent to the respective sphere by virtue of being embedded in the torus surface.

Inversion in a sphere does not change the topology or tangency relationships - it only changes the metrical relationships. QED

Related useful information:

https://hal-univ-bourgogne.archives-ouvertes.fr/hal-00785322/document

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