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
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