SIMON-MARAIS 2020 A4

A regular spatial pentagon consists of five points $P_1, P_2, P_3, P_4, P_5$ in $\mathbb{R}^3$ such that $|P_iP_{i+1}| = |P_jP_{j+1}|$ and $\angle P_{i-1}P_iP_{i+1} = \angle P_{j-1}P_jP_{j+1}$ for all $1 \leq i,j \leq 5$, where $P_0 = P_5$ and $P_6 = P_1$. A regular spatial pentagon is planar if there is a plane passing through all five points $P_1, P_2, P_3, P_4, P_5$.

Show that every regular spatial pentagon is planar.