circle has one center

circle has one center

Theorem 1

.

Every circle has exactly one center.

Before proving this, we state and prove a lemma.

Lemma 1

.

A center of a circle must lie in its interior.

Proof.

Suppose not. Let a circle have a center

O

not lying in its interior. If

O

lies on the circle, then the circle is degenerate (just a point). Suppose that

O

lies in its exterior. Choose a point

A

on the circle such that a line that contains

A

and

O

passes through

the interior of the circle (i.e. (

http://planetmath.org/Ie

), the line is not

tangent

(

http://planetmath.org/TangentLine

) to the circle). Such a line

intersects

the circle at another point

B

. It cannot be the case that

O

is in between

A

and

B

, lest

O

be in the interior of the circle. Without loss of generality, let

B

be between

A

and

O

.

Then

B

⁢

O

<

A

⁢

O

, a

contradiction

, since

O

is supposed to be the center of the circle.
∎

Now to prove the

theorem

.

Proof.

By definition, every circle has at least one center. Suppose that a circle has more than one center. Let

O

and

O

′

be two distinct centers of this circle. By the previous lemma,

O

and

O

′

must lie in the interior of the circle. Draw a chord of the circle which contains both

O

and

O

′

. Let

A

and

B

be the

intersections

of this chord with the circle such that

O

is between (

http://planetmath.org/Betweenness

)

A

and

O

′

. Since

O

and

O

′

are in the interior of the circle, it must be the case that

O

′

is between

O

and

B

.

Note that we must have

A

≠

O

,

A

≠

O

′

,

B

≠

O

, and

B

≠

O

′

as pictured. Otherwise, the circle is degenerate, yielding that

O

=

O

′

. Because of these four

inequalities

, we also have that

A

⁢

O

>

0

,

A

⁢

O

′

>

0

,

B

⁢

O

>

0

, and

B

⁢

O

′

>

0

.

Since

O

is a center of the circle,

A

⁢

O

=

B

⁢

O

. Since

O

′

is a center of the circle,

A

⁢

O

′

=

B

⁢

O

′

. Thus,

A

⁢

O

<

A

⁢

O

′

=

B

⁢

O

′

<

B

⁢

O

, a contradiction. It follows that a circle has exactly one center.
∎

1

Generality

Looking at the proof of the main theorem and the lemma, we see that they hold in any

geometry

in which any two points lie on a common line and in which the notion of betweenness is well-defined. Examples of such geometries include

Euclidean geometry

,

hyperbolic geometry

, and

neutral geometry

. In fact, the uniqueness of center of a circle holds in any

ordered geometry

satisfying the

congruence axioms

(see here (

http://planetmath.org/ProofOfUniquenessOfCenterOfACircle

) for a proof). By contrast, in

spherical geometry

, where the notion of betweenness is not well-defined and a pair of

antipodal points

determines an

infinity

of lines, we also have that every circle has exactly two centers which are antipodal points. However, in

projective geometry

, it is again valid because we identify antipodal points to construct the

projective plane

from the sphere.

More generally, in Riemannian spaces (and

generalizations

such as Finsler spaces), this theorem will hold provided that

geodesics

emanating at a point do not focus at some other point, although they can focus at the same point at which they started. This generalizes what we saw in the case of the sphere and the the projective plane. In both those cases, geodesics focussed but, in the former case, they focussed at the antipodal point but, in the latter case, they only focussed back at their starting point. While on spheres, circles have two centers, we can have Riemannian spaces in which geodesics refocus any number of times, even infinitely often, in which cases a circle could have any number of centers, even infinitely many of them. Because, by geodesic deviation, focussing requires positive

curvature

, we can assert that circles in spaces of non-positive curvature will have unique centers; for

instance

, this explains why the result holds in hyperbolic geometry (which has

constant

negative curvature).

Title

circle has one center

Canonical name

CircleHasOneCenter

Date of creation

2013-03-22 17:15:20

Last modified on

2013-03-22 17:15:20

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

23

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 51M10

Classification

msc 51M04

Classification

msc 51A05

Related topic

RegularPolygonAndCircles

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