KöMaL 2005 B3815

A plane intersects each line segment of a closed polygon $P_1 P_2$...$P_n P_1$ of the space. The intersection with the line segment $P_i P_{i+1}$ is the interior point $Q_i$. Prove that $\frac{P_1Q_1}{Q_1P_2}\cdot\frac{P_2Q_2}{Q_2P_3}\cdot\ldots\cdot \frac{P_{n-1}Q_{n-1}}{Q_{n-1}P_n}\cdot\frac{P_nQ_n}{Q_nP_1}=1.$