How to Apply Euler Formula in this problem I have this problem
Let $L$ be a finite set of lines in the plane, no two of which are parallel and
not all of which are concurrent. Using Euler’s Formula, show that some
point is the point of intersection of precisely two lines of $L$.
I have no idea how to begin, what I think is that I must form an appropriate planar graph, but I don't see it. Can anyone help me?
 A: Take the intersection points of your drawing to be vertices of a graph and the line segments between the intersection points to be the edges.
Now you have a nice planar graph, except for the nasty halflines running to infinity. Fortunately there are only a finite number of intersection points,
so you can walk around your graph, far enough away to enclose all vertices.
During this walk you gather all those halflines running to infinity, cut them off and connect them all to the same (new) vertex (i.e. you "add a vertex at infinity").
The result is still a planar graph, a proper one this time.
Note that a vertex only exists if two lines intersect, which immediately gives the vertex degree 4. Every additional line through such vertex increases the degree by 2.
Now assume that there is no vertex where exactly two lines intersect. This implies that every vertex has degree at least 6. Even the vertex at infinity (why?).
And now we have a contradiction, since a planar graph has minimum degree less than 6.
This last result is where you use Euler's formula. It is rather standard, and easy to prove, but leave a comment if you don't know how, then I will add an explanation.
