Hey math experts, I need a little help with a proof.

I found a sketch of the proof I want to do on the web(attached) , and I find it difficult to understand.

1. The definition of $\displaystyle E$. (I spouse there is a mistake there and it should be $\displaystyle E:=${{$\displaystyle l \in L$ : $\displaystyle l \cap e \not= \emptyset$ } : $\displaystyle e$ is segment}) what do they mean by "e is a segment"?

2. What is the explanation to the existence of such a "$\displaystyle 2/3r$-net" of $\displaystyle \Sigma$?

3. Why does the fact that R is a $\displaystyle 2/3r$-net means that each triangle does not intersect more than $\displaystyle n/r$ lines?

Any help will be appreciated.