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
Curvature of a measure points "x", "y" and "z" in R, let "R"("x", "y", "z") be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear. The Menger curvature "c"("x", "y", "z") is defined to be with the natural convention that "c"("x", "y", "z") = 0 if "x", "y" and "z" are collinear. It is also conventional to extend this definition by setting "c"("x", "y", "z") = 0 if any of the points "x", "y" and "z" coincide. The Menger-Melnikov curvature "c"("μ") of "μ" is defined to be More generally, for "α" ≥ 0, define "c"("μ")