Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
Power of a point centered on the midpoint of O and P and passing through both points. It can also be shown that the point Q is the inverse of P with respect to the given circle. The "power of a point theorem", due to Jakob Steiner, states that for any line through "A" intersecting a circle "c" in points "P" and "Q", the power of the point with respect to the circle "c" is given up to a sign by the product of the lengths of the segments from "A" to "P" and "A" to "Q", with a positive sign if "A" is