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Circle Power


PowerCircle

The power of a fixed point A with respect to a circle of radius r and center O is defined by the product

 p=AP×AQ,
(1)

where P and Q are the intersections of a line through A with the circle. The term "power" was first used in this way by Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967, p. 30). Amazingly, p (sometimes written k^2) is independent of the choice of the line APQ (Coxeter 1969, p. 81).

PowerExternal

Now consider a point P not necessarily on the circumference of the circle. If d=OP is the distance between P and the circle's center O, then the power of the point P relative to the circle is

 p=d^2-r^2.
(2)

If P is outside the circle, its power is positive and equal to the square of the length of the segment PQ from P to the tangent Q to the circle through P,

 p=PQ^2=d^2-r^2.
(3)

If OP lies along the x-axis, then the angle theta around the circle at which Q lies is given by solving

 [(d-costheta)^2+sin^2theta]+1=d^2
(4)

for theta, giving

 theta=+/-sec^(-1)(d/r)
(5)

for coordinates

 (x,y)=r(+/-1/d,sqrt((d^2-1)/(d^2))).
(6)

The points P and P^' are inverse points, also called polar reciprocals, with respect to the inversion circle if

 OP·OP^'=OQ^2=r^2
(7)

(Wenninger 1983, p. 2).

If P is inside the circle, then the power is negative and equal to the product of the diameters through P.

The powers of circle of radius rho with center having trilinear coordinates alpha:beta:gamma with respect to the vertices of a reference triangle are

p_A=(b^2c^2(beta^2+gamma^2+2betagammacosA))/((aalpha+bbeta+cgamma)^2)-rho^2
(8)
p_B=(a^2c^2(alpha^2+gamma^2+2alphagammacosB))/((aalpha+bbeta+cgamma)^2)-rho^2
(9)
p_C=(a^2b^2(alpha^2+beta^2+2alphabetacosC))/((aalpha+bbeta+cgamma)^2)-rho^2
(10)

(P. Moses, pers. comm., Jan. 26, 2005). The circle function of such a circle is then given by

 l=-(p_A)/(bc).
(11)

The locus of points having power k with regard to a fixed circle of radius r is a concentric circle of radius sqrt(r^2+k). 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).


See also

Chord, Chordal Theorem, Circle Function, Coaxal Circles, Inverse Points, Inversion Circle, Inversion Radius, Inversive Distance, Radical Line, Triangle Power

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References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M. and Greitzer, S. L. "The Power of a Point with Respect to a Circle." §2.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 27-31, 1967.Darboux, J. "Mémoir sur les Surfaces Cyclides." Ann. l'École Normale sup. 1, 273-292, 1872.Dixon, R. Mathographics. New York: Dover, p. 68, 1991.Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 153, 1965.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 28-34, 1929.Jurgensen, R. C.; Donnelly, A. J.; and Dolciani, M. P. Th. 42 in Modern Geometry: Structure and Method. Boston, MA: Houghton-Mifflin, 1963.Lachlan, R. "Power of a Point with Respect to a Circle." §300-303 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 183-185, 1893.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. xxii-xxiv, 1995.Steiner, J. "Einige geometrische Betrachtungen." J. reine angew. Math. 1, 161-184, 1826.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, 1983.

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Circle Power

Cite this as:

Weisstein, Eric W. "Circle Power." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CirclePower.html

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