Power of a point P intersects the circle at A and B, the endpoints of the diameter. Their respective products of distances are equal to each other and to the power of point P in that circle This equality is sometimes known as the ""secant-tangent theorem"", ""intersecting chords theorem"", or the ""power-of-a-point theorem"". The power of a point is used in many geometrical definitions and proofs. For example, the radical axis of two given circles is the straight line consisting of points that have equal power to both circles. For each point on this line, there is a unique circle centered on that point
Circle on the unit circle in the complex plane. If "C" is the midpoint of the segment "AB", then the collection of points "P" satisfying the Apollonius condition is not a circle, but rather a line. Thus, if "A", "B", and "C" are given distinct points in the plane, then the locus of points "P" satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius. In every triangle a unique circle, called the incircle, can be inscribed such that it