Given a circle c with center O and radius r, for any point A there is a number called the power of the point A with respect to the circle.
Definition: The power of A with respect to c = |OA|2 - r2
Thought of another way, we can define a Power Function of c, called pc so that pc(A) is the power of A with respect to c. If d is the distance function, with d(A) = |OA|, then:
pc = d2 - r2
or
pc(A) = |OA|2 - r2
Notation Note: Sved uses the notation P(c) for the power of the point P with respect to circle c, but this seems just too different from normal function notation to use.
If A is a point outside the circle, draw a secant intersecting c at M and N, then
pc(A) = |AM||AN|
If A is a point inside the circle, draw a secant intersecting c at M and N, then
pc(A) = - |AM||AN|
If A is a point outside the circle and AT is a line tangent to c at C,
pc(A) = |AT|2
If A is a point outside the circle and e is a circle which is orthogonal to c and centered at A, then if the radius of e is t,
pc(A) = t2
If A is a point and e is a circle centered at A with radius t so that e is orthogonal to two circles c1 and c2, then
Pc1(A) = pc2(A) = t2
Given two circles c1 and c2, the set of points A for which the power of A with respect to c1 and c2 are equal is called the radical axis of c1 and c2.
The radical axis is the locus of centers of circles orthogonal to both c1 and c2.
Theorem: The radical axis of c1 and c2 is a line perpendicular to the line through the centers of the circles.