A line is defined as an infinite set of points extending in both directions. When we have two distinct lines, they can intersect at most at one point. This fundamental property will help us understand why four lines cannot have exactly two intersection points.
Two distinct lines can intersect at most at one point. Given two line equations, we solve the system algebraically to find their intersection. There are three possible cases: parallel lines with no intersection, coincident lines with infinite intersections, or intersecting lines with exactly one intersection point. This fundamental principle is crucial for understanding multi-line configurations.
With four lines, we can have at most six intersection points, calculated as four choose two equals six. These come from all possible pairs of lines. In general position, where no two lines are parallel, we get exactly six intersection points. However, when some lines are parallel, the actual number of intersections decreases.
We can prove that exactly two intersection points is impossible using proof by contradiction. If we assume only two pairs intersect, then the remaining four pairs must be parallel. But this creates an impossible constraint: if L1 is parallel to L2 and L3 is parallel to L4, then L1 cannot simultaneously be parallel to L3. This contradiction proves that exactly two intersection points is impossible for four lines.
Let's examine concrete examples. Four lines can have 0, 1, 3, 4, 5, or 6 intersection points, but never exactly 2. We see all parallel lines give 0 points, three lines through one point give 1 point, three parallel with one crossing give 3 points, and general position gives 6 points. The number 2 is conspicuously absent from all possible configurations, confirming our theoretical proof.