AoPS - Problem

$a$ is side of regular n-gon $A_0A_1...A_{n-1}.$ $P_i \in A_iA_{i+1},$ $Q_i$ is reflection of $P_i$ in midpoint of $A_iA_{i+1},$ all indices are modulo n. $p = \sum_i A_iP_i \cdot A_iP_{i-1}$ $q = \sum_i A_iQ_i \cdot A_iQ_{i-1} = \sum_i (a - A_iP_i) \cdot (a - A_iP_{i-1}) =$ $= \sum_i a^2 - a \sum_i A_iP_i - a \sum_i A_iP_{i-1} + \sum_i A_iP_i \cdot A_iP_{i-1} = \sum_i a^2 - a \sum_i a + p = p$ $[P_0P_1...P_{n-1}] = [A_0A_1...A_{n-1}] - \frac{_p}{^2} \cdot \sin \left(\frac{_{n-2}}{^n} \pi \right) = $ $[A_0A_1...A_{n-1}] - \frac{_q}{^2} \cdot \sin \left(\frac{_{n-2}}{^n} \pi \right)= [Q_0Q_1...Q_{n-1}]$