Formulas relating the number of lines, sections, and intersection points

Some formulas relating the number of intersections, lines, and sections in a plane.

1) The maximum number of intersection points of n lines is . Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2 new intersection points, and so on.

2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the plane. (someone turn this into a formal proof with induction please)

3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point. The proof can be made by bashing the equation with the previous formulas

4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point. Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line.