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
Area of a circle version of the divergence theorem), in a way that avoids all mention of trigonometry and the constant . Consider the vector field formula_27 in the plane. So the divergence of r is equal to two, and hence the area of a disc "D" is equal to By Green's theorem, this is the same as the outward flux of r across the circle bounding "D": where n is the unit normal and "ds" is the arc length measure. For a circle of radius "R" centered at the origin, we have formula_30 and formula_31, so the above equality is The integral of