Let S be a set of n lines in the plane such that no two are parallel and no three meet in the same point. Show, by induction, that the lines in S determine O(n²) intersection points.

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Any Pair Segment

A line segment is the part of a line that is represented by a connection of two end points. It encompasses both ends of the line, as well as all points in between. The length of a line cannot be measured, but the length of a segment can be measured. If a line segment is bounded by two end points, it is called a closed line segment, whereas an open line segment does not have both endpoints. A line segment with a single endpoint is called half-open. A closed line segment contains every point on the line that exists between the two end points. For example, if L is a line, then AB is a line segment of line L where A and B are endpoints of the line segment.