76 circle theorems

from 2 to 3.141592… for different circles drawn on

the surface of a

SPHERE

, for instance, since the diameter

of a circle must be measured as the length of a curved

line on the surface.)

If Cis the circumference of a planar circle and

D= 2ris its diameter, then, by definition, π= C/D.

This yields a formula for the circumference of a circle:

C= 2πr

A study of

AREA

also shows that the area Aof a circle

is given by:

A= πr2

It is not immediate that the value πshould also appear

in this formula.

A study of

EQUIDISTANCE

shows that it is always

possible to draw a circle through any three given points

in a plane (as long as the points do not lie in a straight

line), or, equivalently, it is always possible to draw a

CIRCUMCIRCLE

for any given

TRIANGLE

. A

POLLONIUS OF

P

ERGA

(ca. 262–190

B

.

C

.

E

.) developed general methods

for constructing a circle

TANGENT

to any three objects

in the plane, be they points, lines, or other circles.

Any line connecting two points on a circle is called

a

CHORD

of the circle. It divides the circle into two

regions, each called a segment. A chord of maximal

length passes through the center of a circle and is also

called a diameter of the circle. (Thus the word diameter

is used interchangeably for such a line segment and for

the numerical value of the length of this line segment.)

A radius of a circle is any line segment connecting the

center of the circle to a point on the circle. Two differ-

ent radii determine a wedge-shaped region within the

circle called a sector. If the angle between the two radii

is θ, given in

RADIAN MEASURE

, then the area of this

segment is . The length of the

ARC

of

the circle between these two radii is .

Any two points P= (a1,b1) and Q= (a2,b2) in the

plane determine a circle with the line segment connect-

ing Pto Qas diameter. The equation of this circle is

given by:

(x– a1)(x– a2) + (y– b1)(y– b2) = 0

The J

ORDAN CURVE THEOREM

establishes that a cir-

cle divides the plane into two regions: an inside and an

outside. (This seemingly obvious assertion is not true for

circles drawn on a

TORUS

, for example.) Two intersect-

ing circles divide the plane into four regions; three inter-

secting circles can be arranged to divide the plane into

eight regions; and four mutually intersecting circles can

divide the plane into 14 regions. In general, the maximal

number of regions into which nintersecting circles

divide the plane is given by the formula: n2– n+ 2.

The region formed at the intersection of two inter-

secting circles of the same radius is called a lens.

There are a number of

CIRCLE THEOREMS

describ-

ing the geometric properties of circles. A circle is a

CONIC SECTION

. It can be regarded as an

ELLIPSE

for

which the two foci coincide.

If one permits the use of

COMPLEX NUMBERS

, then

any two circles in the plane can be said to intersect. For

example, the two circles each of radius one centered

about the points (0,0) and (4,0), respectively, given by

the equations x2+ y2= 1 and (x– 4)2+ y2= 1 intersect

at the points (2, i√

–

3) and (2,–i√

–

3).

The three-dimensional analog of a circle is a

SPHERE

: the locus of all points equidistant from a fixed

point Oin three-dimensional space. In one-dimension,

the analog of a circle is any pair of points on a number

line. (Two points on a number line are equidistant from

their

MIDPOINT

.)

The midpoint theorem asserts that all midpoints of

line segments connecting a fixed point Pin the plane to

points on a circle Cform a circle of half the radius of C.

See also A

POLLONIUS

’

S CIRCLE

; B

RAHMAGUPTA

’

S

FORMULA

;

CYCLIC POLYGON

; F

AREY SEQUENCE

;

NINE

-

POINT CIRCLE

;

UNIT CIRCLE

; V

ENN DIAGRAM

.

circle theorems A

CIRCLE

is defined as the set of

points in a plane that lie a fixed distance r, called the

radius, from some fixed point O, called the center. This

simple definition has a number of significant geometric

consequences:

1. Tangent Theorems

The point of contact of a

TANGENT

line with a circle

is the point on that line closest to the center point

O. As a consequence of P

YTHAGORAS

’

S THEOREM

,

the line connecting the point of contact to Ois at

an angle 90°to the tangent line. This proves:

The tangent to a circle is

PERPENDICULAR

to

the radius at the point of contact.