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 point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant "ratio" (other than 1) of distances to two fixed foci, "A" and "B". (The set of points where the distances are equal is the perpendicular bisector of segment "AB", a line.) That circle is sometimes said to be drawn "about" two points. The proof is in two parts. First, one must prove that, given two foci "A" and "B" and a ratio of distances, any point "P" satisfying