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 These eight or fewer triple crossing points are the centers of inversions that take all three circles "α", "β", and "γ" to become equal circles. For three circles that are mutually externally tangent, the (unique) circles of antisimilitude for each pair again cross each other at 120° angles in two triple intersection points that are the isodynamic points of the triangle formed by the three points of tangency. 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