This is a classic problem, in disguise. That problem was open for 20+ years, so if you solve this problem without using the classic result, pat yourself on the back!
You are given a finite set of lines, $S$ in the 2D plane, no two lines of which are parallel and not all are concurrent.
Show that there are two lines (call them $p$ and $q$) such that no other line in $S$ passes through the intersection point of $p$ and $q$.
i.e. among all the points of intersections formed by the lines in $S$, there is a point through which only two lines of $S$ meet.
[Solution]
You are given a finite set of lines, $S$ in the 2D plane, no two lines of which are parallel and not all are concurrent.
Show that there are two lines (call them $p$ and $q$) such that no other line in $S$ passes through the intersection point of $p$ and $q$.
i.e. among all the points of intersections formed by the lines in $S$, there is a point through which only two lines of $S$ meet.
[Solution]
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