Suppose that you have a drawing in which a ring of k circles is tangent to each other, and each circle in the ring is also tangent to two more circles A and B. Then, any other circle C that is tangent to A and B along the same bisector is also part of a similar ring of k circles. This fact is known as Steiner's Porism.
In the animation below, the three black anchors move the two circles A and B; moving the red anchor produces different rings of six circles for the same choice of A and B.
The usual proof of this is simply to choose an inversion that makes A and B concentric, after which the result follows immediately by symmetry.
Any three tangent circles have two inscribed and circumscribed circles forming an instance of Steiner's porism with k=3. When k=4, the six circles of the porism can be grouped in two other ways into instances of the same construction. In this case, the graph describing the tangencies between the six circles is the skeleton of an octahedron.
Animation created by Cinderella.
From the Geometry Junkyard, computational and recreational geometry.
David Eppstein, Theory Group, ICS, UC Irvine.
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