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
Group of rational points on the unit circle to the origin, equal to 0. The identity in this larger group is (1,0,1,0), and the group operation is formula_9 For the group on the unit circle, the appropriate subgroup is points of form ("w","x",1,0), with formula_10 and its identity element is (1,0,1,0). The unit hyperbola group corresponds to points of form (1,0,"y","z"), with formula_11 and the identity is again (1,0,1,0). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) Group of rational points on the unit circle In mathematics, the rational points on the unit circle are those points ("x",