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 by three points formula_10 not on a line is obtained by a conversion of the "3-point-form of a circle's equation" Homogeneous form<br> In homogeneous coordinates, each conic section with the equation of a circle has the form It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points "I"(1: "i": 0) and "J"(1: −"i": 0). These points are called the circular points at infinity. In polar coordinates, the equation of a circle is: where "a" is the radius of the circle, formula_14 is the polar coordinate of