Steiner's Porism


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For two concentric circles, either there exists a closed chain of circles tangent to the given two as well as to their immediate neighbors in the chain, or such a chain does not exist. In the former case, the chain could be started at the arbitrary location in the ring between the two circles.

Jacob Steiner's wonderful theorem says that the same holds true even if the two circles are not concentric. A simple proof depends on the following assertion:

Any two non-intersecting circles can be inverted into concentric circles.

Let first two circles ∑1 and ∑2 lie outside each other. Their radical axis consists of the points from which the tangents to ∑1 and ∑2 are equal. A circle centered on the radical axis and having radius equal to the common tangent to the circles ∑1 and ∑2 from its center is perpendicular to both circles. It is therefore easy to find two intersecting circles α1 and α2 orthogonal to ∑1 and ∑2. Make an inversion with the center at one of the points of intersection of α1 and α2. α1 and α2 will map onto two straight intersecting lines. Let T' be their point of intersection. ∑1 and ∑2 will map onto two circles orthogonal to those lines and therefore both centered at the point T'. (We can prove even a stronger result.)

If one of the given circles is located in the interior of the other, we may first make an inversion that will separate the two. Any inversion with the center in the ring formed by the two circles will serve that purpose.

A different illustration of Steiner's porism is available elsewhere.

References

  1. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
  2. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
  3. H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967

Inversion - Introduction

Radical Axis and Radical Center

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