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N lines cutting each other in the plane

Source: Iranian Math Olympiad(Second Round 2016)

May 5, 2016
combinatorics

Problem Statement

Let l1,l2,l3,...,Lnl_1,l_2,l_3,...,L_n be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let AA be the intersection point of lines li,ljl_i,l_j. We call AA an "Interior Point" if there are points C,DC,D on lil_i and E,FE,F on ljl_j such that AA is between C,DC,D and E,FE,F. Prove that there are at least (n−2)(n−3)2\frac{(n-2)(n-3)}{2} Interior points.(n>2n>2) note: by point here we mean the points which are intersection point of two of l1,l2,...,lnl_1,l_2,...,l_n.
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