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
Dividing a circle into areas cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact. Lemma. We can choose the new point "A" so that case "b" occurs for each of the new lines. Proof. Notice that, for the case "a", three points must be on one line: the new point "A", the old point "O" to which we draw the line, and the point "I" where two of the old lines intersect. Notice that there are "n" old points "O", and hence finitely many points "I"