Question 19
Let $n$ be a positive integer. Points $P_1, P_2, \dots, P_{4n}$ are placed in a plane in such a way that no 3 points among them lie on any straight line. Furthermore, for each $i = 1, 2, \dots, 4n$ if we rotate the half-line $P_i P_{i-1}$ starting at $P_i$ around the point $P_i$ by $90^\circ$ clockwise, then the half line falls onto the half-line $P_i P_{i+1}$ starting at $P_i$. Determine the maximum possible number of the pairs $(i, j)$ for which the line segments $P_i P_{i+1}$ and $P_j P_{j+1}$ intersect at a point different from the end points of the line segments. Here we let $P_0 = P_{4n}, P_{4n+1} = P_1$ and assume that $1 \le i \lt j \le 4n$.