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 Let "S" be a sphere with center "O", "P" a plane which intersects "S". Draw perpendicular to "P" and meeting "P" at "E". Let "A" and "B" be any two different points in the intersection. Then "AOE" and "BOE" are right triangles with a common side, "OE", and hypotenuses "AO" and "BO" equal. Therefore, the remaining sides "AE" and "BE" are equal. This proves that all points in the intersection are the same distance from the point "E" in the plane "P", in other words all points in the intersection lie on a circle "C" with center "E". This proves