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 a sphere there is exactly one circle that can be drawn through three given points. The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles. To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius formula_1) is centered at the origin. Points on this sphere satisfy Also without loss of generality, assume that the second sphere, with radius formula_3, is centered at a point on the positive x-axis, at distance formula_4 from the origin. Its points