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 and ellipses work)+ 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.