## Power (Circle)

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

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

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 . If is inside the Circle, then the Power is Negative and equal to the product of the Diameters through .

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).

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.