Steiner's Porism

Suppose that a circle C lies inside circle D,
and that circle E touches both of them.
We can draw a sequence of circles, each touching C and D
and the previous member of the sequence.

There are two possibilities, either
the sequence finishes, with the last circle touching E (upper picture),
or the last circle cuts E (lower picture).
In the first case, we call the sequence a Steiner Chain.

Steiner's Porism
If we have a Steiner Chain for one choice of E,
then any other choice of E results in a Steiner Chain
with the same number of circles.

The proof is quite simple.

Note that C and D are non-intersecting i-lines.
By the Concentric Circles Theorem, we can invert the figure so that
the images of C and D are concentric.
The starting circle will invert into an i-line touching both, i.e. to a circle.

Since the figure is symmetric about the common centre P,
it is clear that, if starting with E gives a Steiner Chain,
then starting with F will give the Steiner Chain obtained by rotation.
Clearly, it will contain the same number of circles.

The CabriJava pane on the right shows a typical Steiner Chain
(the red circles are non-concentric).
If you move your mouse over the pane,
you should see the chain rotating about the inner circle.

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