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 of antisimilitude These eight or fewer triple crossing points are the centers of inversions that take all three circles "α", "β", and "γ" to become equal circles. For three circles that are mutually externally tangent, the (unique) circles of antisimilitude for each pair again cross each other at 120° angles in two triple intersection points that are the isodynamic points of the triangle formed by the three points of tangency. Circle of antisimilitude In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, "α" and "β", is a reference circle for which "α" and "β" are inverses of