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
Distance from a point to a line The line through these two points is perpendicular to the original line, so Thus, formula_7 and by squaring this equation we obtain: Now consider, using the above squared equation. But we also have, since ("m", "n") is on "ax" + "by" + "c" = 0. Thus, and we obtain the length of the line segment determined by these two points, This proof is valid only if the line is not horizontal or vertical. Drop a perpendicular from the point "P" with coordinates ("x", "y") to the line with equation "Ax" + "By" + "C" = 0. Label the foot of