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
- J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
- H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
- H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967
Inversion - Introduction
- Angle Preservation Property
- Apollonian Circles Theorem
- Archimedes' Twin Circles and a Brother
- Bisectal Circle
- Chain of Inscribed Circles
- Circle Inscribed in a Circular Segment
- Circle Inversion: Reflection in a Circle
- Circle Inversion Tool
- Feuerbach's Theorem: a Proof
- Four Touching Circles
- Hart's Inversor
- Inversion in the Incircle
- Inversion with a Negative Power
- Miquel's Theorem for Circles
- Peaucellier Linkage
- Polar Circle
- Poles and Polars
- Ptolemy by Inversion
- Radical Axis of Circles Inscribed in a Circular Segment
- Steiner's porism
- Stereographic Projection and Inversion
- Tangent Circles and an Isosceles Triangle
- Tangent Circles and an Isosceles Triangle II
- Three Tangents, Three Secants
- Viviani by Inversion
- Simultaneous Diameters in Concurrent Circles
- An Euclidean Construction with Inversion
- Construction and Properties of Mixtilinear Incircles
- Two Quadruplets of Concyclic Points
- Seven and the Eighth Circle Theorem
- Invert Two Circles Into Equal Ones
Radical Axis and Radical Center
- How to Construct a Radical Axis
- A Property of the Line IO: A Proof From The Book
- Cherchez le quadrilatere cyclique II
- Circles On Cevians
- Circles And Parallels
- Circles through the Orthocenter
- Coaxal Circles Theorem
- Isosceles on the Sides of a Triangle
- Properties of the Circle of Similitude
- Six Concyclic Points
- Radical Axis and Center, an Application
- Radical axis of two circles
- Radical Axis of Circles Inscribed in a Circular Segment
- Radical Center
- Radical center of three circles
- Steiner's porism
- Stereographic Projection and Inversion
- Stereographic Projection and Radical Axes
- Tangent as a Radical Axis
- Two Circles on a Side of a Triangle
- Pinning Butterfly on Radical Axes
- Two Lines - Two Circles
- Two Triples of Concurrent Circles
- Circle Centers on Radical Axes
- Collinearity with the Orthocenter
- Six Circles with Concurrent Pairwise Radical Axes
- Six Concyclic Points on Sides of a Triangle
- Line Through a Center of Similarity
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