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 C is the circumference of a planar circle and
D = 2r is 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 A of a circle
is given by:
A = πr
2
It is not immediate that the value π should 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. APOLLONIUS OF
PERGA (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 different 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 = (a 1 ,b 1 ) and Q = (a 2 ,b 2 ) in the
plane determine a circle with the line segment connecting P to Q as diameter. The equation of this circle is
given by:
(x – a 1 )(x – a 2 ) + (y – b 1 )(y – b 2 ) = 0
The JORDAN CURVE THEOREM establishes that a circle 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 intersecting circles divide the plane into four regions; three intersecting 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 n intersecting circles
divide the plane is given by the formula: n
2 – n + 2.
The region formed at the intersection of two intersecting circles of the same radius is called a lens.
There are a number of CIRCLE THEOREMS describing 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 x
2 + y
2 = 1 and (x – 4)
2 + y
2 = 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 O in 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 P in the plane to
points on a circle C form a circle of half the radius of C.
See also APOLLONIUS’S CIRCLE; BRAHMAGUPTA’S
FORMULA; CYCLIC POLYGON; FAREY SEQUENCE; NINEPOINT CIRCLE; UNIT CIRCLE; VENN 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 PYTHAGORAS’S THEOREM,
the line connecting the point of contact to O is 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.
θ
π
π
θ
2
2
⋅
=
r r
θ
π
π
θ
2
1
2
2
2
⋅
=
r
r
76 circle theorems
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 C is the circumference of a planar circle and
D = 2r is 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 A of a circle
is given by:
A = πr
2
It is not immediate that the value π should 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. APOLLONIUS OF
PERGA (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 different 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 = (a 1 ,b 1 ) and Q = (a 2 ,b 2 ) in the
plane determine a circle with the line segment connecting P to Q as diameter. The equation of this circle is
given by:
(x – a 1 )(x – a 2 ) + (y – b 1 )(y – b 2 ) = 0
The JORDAN CURVE THEOREM establishes that a circle 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 intersecting circles divide the plane into four regions; three intersecting 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 n intersecting circles
divide the plane is given by the formula: n
2 – n + 2.
The region formed at the intersection of two intersecting circles of the same radius is called a lens.
There are a number of CIRCLE THEOREMS describing 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 x
2 + y
2 = 1 and (x – 4)
2 + y
2 = 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 O in 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 P in the plane to
points on a circle C form a circle of half the radius of C.
See also APOLLONIUS’S CIRCLE; BRAHMAGUPTA’S
FORMULA; CYCLIC POLYGON; FAREY SEQUENCE; NINEPOINT CIRCLE; UNIT CIRCLE; VENN 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 PYTHAGORAS’S THEOREM,
the line connecting the point of contact to O is 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.
θ
π
π
θ
2
2
⋅
=
r r
θ
π
π
θ
2
1
2
2
2
⋅
=
r
r
76 circle theorems