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
Five points determine a conic justify. Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 2 = 8. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7. Five points determine a conic In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a