Steiners Alternative:

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Steiners Alternative An Introduction to
Inversive Geometry
Bruce Cohen Lowell High School,
SFUSD bic_at_cgl.ucsf.edu http//www.cgl.ucsf.edu/hom
e/bic
David Sklar San Francisco State
University dsklar_at_sfsu.edu
Asilomar - December 2005
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Plan
Discovering Steiners Alternative
Handout Statement of the theorem
Sketch of the proof
A step beyond the basics The Reduction of
Two Circles Concepts in the proof
Power, Radical Axis, Coaxial Pencil, Limit
Point Completing the proof
Basics of Inversive Geometry Inversion in
a circle Lines go to circles or lines
Circles go to circles or lines Angles
are preserved A very brief history

Where could we go from here? Four
possible applications Where cant we go from
here? The Great Poncelet Theorem
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Part I Discovering Steiners Alternative
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(No Transcript)
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Steiners Alternative (or Steiners Porism)
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Porism a finding of conditions that render an
existing theorem indeterminate or capable of many
solutions. -- Steven Schwartzman, The
Words of Mathematics
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A Sketch of the Proof of Steiners Alternative
Given two nonintersecting circles there exists a
continuous, invertible, circle preserving
transformation from the plane to itself that
maps the given non-intersecting circles to
concentric circles. Letting T denote such a
transformation (a specially chosen inversion in
a circle) we have
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Part II Basics of Inversive Geometry
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Inversion in a Circle
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Lines go to Circles or Lines
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Circles go to Circles or Lines
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Angles are Preserved
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Summary Properties of Inversion
Points inside the circle of inversion go to
points outside, points outside go to points
inside, points on the circle are fixed and, like
reflection, the transformation is self inverse
Inversion preserves the family of circles and
lines. Specifically
Circles that dont pass through the center of the
circle of inversion are mapped to circles that
dont pass through the inversion center (but
inversion does not send centers to centers)
Circles that pass through the center of the
circle of inversion are mapped to lines that
dont pass through the inversion center
Lines that dont pass through the center of the
circle of inversion are mapped to circles that
pass through the inversion center
Lines that pass through the center of the circle
of inversion are mapped to themselves (although
their points are not fixed points)
Inversion is an angle preserving map, like
reflection, the angle between the tangent lines
of two intersecting curves is the same as the
angle between the tangent lines of their image
curves
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A Brief History of Inversive Geometry
The idea of inversion is ancient, and was used by
Apollonius of Perga about 200 BC.
The invention of Inversive Geometry is usually
credited to Jakob Steiner whose work in the
1820s showed a deep understanding of the
subject.
The first explicit description of inversion as a
transformation of the punctured plane was
presented by Julius Plücker in 1831.
The first comprehensive geometric theory is due
to August F. Möbius in 1855.
The first modern synthetic-axiomatic construction
of the subject is due to Mario Pieri in 1910.
-- Source Jim Smith
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Part III A Step Beyond the Basics
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The Reduction of Two Circles Theorem
The proof is (really) constructive. We will show
how to find by a compass and straight-edge
construction, from the given circles, two points
such that inversion in a circle centered at
either point sends the given circles to
concentric circles. To help understand why the
construction works its useful to introduce some
interesting, and perhaps unfamiliar, concepts
about circles. These concepts are power, radical
axis, pencil, and limit point.
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The Power of a Point with Respect to a Circle
The power of a point A outside of the circle is
positive and equal to the square of the distance
from A to the point of tangency B.
The power of a point on the circle is zero.
The power of a point A inside of the circle is
negative and equal to the negative of the square
of the distance from A to the point where the
chord perpendicular to the radius through A
intersects the circle.
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The Radical Axis of Two Non-Concentric Circles
The locus of points that have the same power with
respect to two non-concentric circles is a line
perpendicular to their line of centers.
Proof Without loss of generality introduce a
coordinate system with the x-axis as the line of
centers, the origin at the center of one circle
and the center of the other at the point (h, 0).
a line perpendicular to the line of centers
The locus of points that have the same power with
respect to two non-concentric circles is called
the Radical Axis of the two circles.
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Radical Axes Examples
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Constructing the Radical Axis of Two
Non-intersecting Circles
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Pencils of Coaxial Circles
The Pencil of Circles determined by two
non-concentric circles C and D is the set of all
circles whose centers lie on their line of
centers, and such that the radical axis of any
pair of circles in the set is the same as the
radical axis of C and D.
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Limit Points of Pencils of Non-intersecting
Coaxial Circles
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Proof of the Reduction of Two Circles Theorem
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Part IV Where Could We Go from Here?
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Four Possibilities
A more quantitative development of inversive
geometry including the concept of the inversive
distance between two circles. This would allow
the use of a quick computation to tell whether a
Steiner chain is finite.
William Thomson (Lord Kelvin) used inversion to
compute the effect of a point charge on a nearby
conductor consisting of two intersecting planes
An application of pencils of nonintersecting
circles in the study of the three-sphere
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From Marcel Bergers Geometry II
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Part V Where Cant We Go from Here?
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Poncelets Alternative The Great Poncelet
Theorem for Circles
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Bibliography
1. M. Berger, Geometry I and Geometry II,
Springer-Verlag, New York, 1987

• H.S.M. Coxeter S.L. Greitzer, Geometry
  Revisited, The Mathematical
• Association of America, Washington, D.C.,
  1967

3. I. J. Schoenberg, On Jacobi-Bertrands
Proof of a Theorem of Poncelet, in
Studies in Pure Mathematics to the Memory of Paul
Turán (xxx edition), Hungarian Academy of
Sciences, Budapest, pages 623-627.
4. C.S. Ogilvy, Excursions in Geometry, Dover,
New York, Dover 1990
5. S.Schwartzman, The Words of Mathematics,
The Mathematical Association of America,
Washington, D.C., 1994
6. J.T. Smith E.A. Marchisotto, The Legacy
of Mario Pieri in Geometry and Arithmetic,
Manuscript (email smith_at_math.sfsu.edu for access)
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The Concentric Case
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Warm-up Problem 1 (b)
The locus of centers of circles tangent to
circles C and D is an ellipse with foci at
the centers of C and D such that the sum of
the distance to the foci is the sum of the radii
of C and D.