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 also orthogonal to both circles of antisimilitude. Suppose that, for three circles "α", "β", and "γ", there is a circle of antisimilitude for the pair ("α","β") that crosses a second circle of antisimilitude for the pair ("β","γ"). Then there is a third circle of antisimiltude for the third pair ("α","γ") such that the three circles of antisimilitude cross each other in two triple intersection points. Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross.