[alogo] Steiner Chain

These are chains of circles (sequence a0, a1, ...) tangent each to the previous and all to two circles c1, c2.
Here we discuss the theorem of Steiner about finite chains of circles contained between two circles c1, c2 one of which is entirely inside the other.
Theorem If the last circle an of the chain intersects/contacts/not-intersects the first one a0, this will be true also for any position of the initial circle a0.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

The theorem follows at once by inverting the whole system of c1, c2 and the chain a0, a1, ... with respect to an appropriate circle c. Circle c has its center A on one of the two limit points of the circle bundle (I), consisting of circles orthogonal to both c1 and c2. The radius of c can be taken arbitrary (here controlled through point F). The inversion F with respect to c has the property to transform the two non intersecting circles to two concentric d1, d2. The whole chain a0, a1, ... is then transformed via F to a chain of equal circles b0, b1, ... lying between d1, d2. The theorem follows immediately from the rotational symmetry of the inverted system d1, d2, b0, b1, ... .
The figure above was constructed the inverse way: First the chain of circles b0, b1, ... through repeated rotation of b0 about E by the angle w (the angle of tangents to b0 from E), then by inverting to circles a0, a1, ... .

See Also



Steiner, J. Werke Bd. I, II. New York, Chelsea, 1971, bd I, p. 43.

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