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Power (Circle)

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The Power of the two points $P$ and $Q$ with respect to a Circle is defined by

p\equiv OP\times PQ.

Let $R$ be the Radius of a Circle and $d$ be the distance between a point $P$ and the circle's center. Then the Power of the point $P$ relative to the circle is


If $P$ is outside the Circle, its Power is Positive and equal to the square of the length of the segment from $P$ to the tangent to the Circle through $P$. If $P$ is inside the Circle, then the Power is Negative and equal to the product of the Diameters through $P$.

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 Chordal Theorem, Coaxal Circles, Inverse Points, Inversion Circle, Inversion Radius, Inversive Distance, Radical Line


Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 27-31, 1967.

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.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. xxii-xxiv, 1995.

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© 1996-9 Eric W. Weisstein