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 antisimilitude Circle of antisimilitude In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, "α" and "β", is a reference circle for which "α" and "β" are inverses of each other. If "α" and "β" are non-intersecting or tangent, a single circle of antisimilitude exists; if "α" and "β" intersect at two points, there are two circles of antisimilitude. When "α" and "β" are congruent, the circle of antisimilitude degenerates to a line of symmetry through which "α" and "β" are reflections of each other. If the two circles "α" and "β" cross each other, another two