Steiner’s Alternative:
An Introduction to Inversive
Geometry
Bruce Cohen
Lowell High School, SFUSD
bic@cgl.ucsf.edu
http://www.cgl.ucsf.edu/home/bic
David Sklar
San Francisco State University
dsklar@sfsu.edu
Asilomar - December 2005
Plan
Discovering Steiner’s Alternative
A step beyond the basics
Handout
The Reduction of Two Circles
Statement of the theorem
Concepts in the proof
Sketch of 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
Power, Radical Axis,
Coaxial Pencil, Limit Point
Completing the proof
Where could we go from here?
Four possible applications
Where can’t we go from here?
The Great Poncelet Theorem
Part I
Discovering Steiner’s Alternative
1
2
C
D
Steiner’s Alternative (or Steiner’s Porism)
Let C and D be two circles, with C inside D, and let  1 be a circle externally
tangent to C and internally tangent to D. Then construct a chain of circles
 1 ,  2 , . . . ,  i , . . . , determined by  1 , such that for each i let  i 1 be tangent
to  i , C , D, and, for i  2 distinct from  i 1. Steiner's Alternative says that
if  n   1 for some n  1, then for any other initial circle  1 we will have  n   1.
7
 7
1
 1
C
C
D
D
Note: It follows that if  i   1 for all i  1, then for any other initial circle  1 we
will have  i   1.
Porism: … a finding of conditions that render an existing
theorem indeterminate or capable of many solutions.
-- Steven Schwartzman, The Words of Mathematics
A Sketch of the Proof of Steiner’s 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
T
R
T 1
T
Part II
Basics of Inversive Geometry
Inversion in a Circle
Lines go to Circles or Lines
Circles go to Circles or Lines
Angles are Preserved
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 don’t pass through the center of the circle of inversion are
mapped to circles that don’t 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 don’t pass through the inversion center
Lines that don’t 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
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 1820’s 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
“Jakob Steiner’s mathematical
work was confined to geometry.
This he treated synthetically, to
the total exclusion of analysis,
which he hated, and he is said to
have considered it a disgrace to
synthetical geometry if equal or
higher results were obtained by
analytical methods.”
-- Source: Wikipedia
Part III
A Step Beyond the Basics
The Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
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 it’s useful to introduce some interesting, and
perhaps unfamiliar, concepts about circles. These concepts are
power, radical axis, pencil, and limit point.
The Power of a Point with Respect to a Circle
If  is a circle of radius r and A is a point at distance d from the
center of  then the power of A with respect to  is d 2  r 2
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.
B
A2
A1
r
r
d
d
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.

A3
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( x, y )
Let A( x, y ) be a point that has the same
power with respect to each circle, then
d12  r12  d 2 2  r2 2
2
2
1
2
d2
r2
y
r1
x  y  r  ( x  h)  y  r2
2
d1
2
2
x
( h,0)
h 2  (r2 2  r12 )
x
2h
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.
Radical Axes Examples
Constructing the Radical Axis of Two Non-intersecting Circles
Draw the line of centers of circles C and D.
Draw a circle E that intersects C and D
whose center is not on their line of centers.
E
P
Draw L1 , the radical axis of circles E and C.
L2
Draw L 2 , the radical axis of circles E and D.
L1 and L 2 intersect at a point P that has
the same power with respect to each of
the E , C , and D.
C
L1
D
Since P has the same power with respect to C and D it lies on their radical axis,
so the line through P perpendicular to their line of centers is the radical axis
of C and D.
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.
Intersecting Pencil
C
D
Non-Intersecting Pencil
C
D
Limit Points of Pencils of Non-intersecting Coaxial Circles
C
D
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Proof of the Reduction of Two Circles Theorem
Theorem Two non-intersecting circles C and D can always be
transformed, by an inversion, into two concentric circles
C and D.
Part IV
Where Could We Go from Here?
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
Higher dimensional inversive geometry: T (v ) 
r2
v
2
v
An application of pencils of nonintersecting circles in
the study of the three-sphere
From Marcel Berger’s
Geometry II
Part V
Where Can’t We Go from Here?
“Poncelet’s Alternative”: The Great Poncelet Theorem for Circles
Let C and D be two circles, with D inside D. Construct a sequence of points
P1 , P2 , . . . , Pi , . . . on D, such that for each i the line segment PP
i i 1 is tangent
to C and (for i  2) distinct from PP
i i 1 . "Poncelet's Alternative" says that
if Pn  P1 for some n  1, then for any other initial point P1 we will have Pn  P1.
Despite the similarity in the statements of the
two theorems, Poncelet's theorem remains
much more difficult to prove than Steiner's.
Bibliography
1. M. Berger, Geometry I and Geometry II, Springer-Verlag, New York, 1987
2.
H.S.M. Coxeter & S.L. Greitzer, Geometry Revisited, The Mathematical
Association of America, Washington, D.C., 1967
3. I. J. Schoenberg, “On Jacobi-Bertrand’s 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@math.sfsu.edu for access)
The Concentric Case

sin  
Rr
2

r


Rr
Rr
R 1  sin 

r 1  sin 
r 
r
R
The chain will close after one circuit if and

only if  
for some integer n  3.
n
It will close after k circuits if and only if  

k
n
, with
 3.
n
k
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.
PA  rC  r
PB  rD  r
PA  PB  rC  rD
r
P
r
rD
rC
A
B
C
D