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 on the point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant "ratio" (other than 1) of distances to two fixed foci, "A" and "B". (The set of points where the distances are equal is the perpendicular bisector of segment "AB", a line.) That circle is sometimes said to be drawn "about" two points. The proof is in two parts. First, one must prove that, given two foci "A" and "B" and a ratio of distances, any point "P" satisfying