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 where two of the old lines intersect. For each "O" and "I", the line "OI" crosses the circle in one point other than "O". Since the circle has infinitely many points, it has a point "A" which will be on none of the lines "OI". Then, for this point "A" and all of the old points "O", case "b" will be true. This lemma means that, if there are "k" lines crossing "AO", then each of them crosses "AO" at a different point and "k+1" new areas are created by the line "AO". The lemma establishes an important property for