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 also orthogonal to both circles of antisimilitude. Suppose that, for three circles "α", "β", and "γ", there is a circle of antisimilitude for the pair ("α","β") that crosses a second circle of antisimilitude for the pair ("β","γ"). Then there is a third circle of antisimiltude for the third pair ("α","γ") such that the three circles of antisimilitude cross each other in two triple intersection points. Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross.