Power (Circle)

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

$$p=d^2-R^2.$$

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

References

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.