The power of a fixed point 
with respect to a circle of radius 
and center 
is defined by the product
 |
(1)
|
where 
and 
are the intersections of a line through
with the circle. The term "power"
was first used in this way by Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967,
p. 30). Amazingly, 
(sometimes written 
)
is independent of the choice of the line 
(Coxeter 1969, p. 81).
Now consider a point 
not necessarily on the circumference of the circle. If 
is the distance between 
and the circle's center 
, then the power of the point 
relative to the circle is
 |
(2)
|
If 
is outside the circle,
its power is positive and equal to the square of the
length of the segment 
from 
to the tangent 
to the circle through 
,
 |
(3)
|
If 
lies along the x-axis,
then the angle 
around the circle at which 
lies is given by solving
![[(d-costheta)^2+sin^2theta]+1=d^2](/images/equations/CirclePower/NumberedEquation4.svg) |
(4)
|
for 
, giving
 |
(5)
|
for coordinates
 |
(6)
|
The points 
and 
are inverse
points, also called polar reciprocals, with respect to the inversion
circle if
 |
(7)
|
(Wenninger 1983, p. 2).
If 
is inside the circle,
then the power is negative and equal to the product
of the diameters through 
.
The powers of circle of radius 
with center having trilinear coordinates 
with respect to the vertices of a reference
triangle are
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
(P. Moses, pers. comm., Jan. 26, 2005). The circle function of such a circle is then given by
 |
(11)
|
The locus of points having power 
with regard to a fixed circle
of radius 
is a concentric circle
of radius 
. The chordal theorem
states that the locus of points having equal power
with respect to two given nonconcentric circles is a line
called the radical line (or chordal; Dörrie
1965).
