Question 20
Let $P_1, P_2, \dots, P_{2n}$ be a permutation of the vertices of a regular $2n$-gon. Prove that every closed polygonal line that consists of segments
$$
\overline{P_1P_2}, \overline{P_2P_3}, \dots, \overline{P_{2n-1}P_{2n}}, \overline{P_{2n}P_1}
$$
contains at least one pair of parallel segments.
$$
\overline{P_1P_2}, \overline{P_2P_3}, \dots, \overline{P_{2n-1}P_{2n}}, \overline{P_{2n}P_1}
$$
contains at least one pair of parallel segments.