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
Dividing a circle into areas Dividing a circle into areas In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with "n" sides, in such a way as to "maximise" the number of areas created by the edges and diagonals, has a solution by an inductive method. If we already have "n" points on the circle and add one more point, we draw "n" lines from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points)