ILL 1983 Problem 34

In a plane are given n points $P_i \ (i = 1, 2, \ldots , n)$ and two angles $\alpha$ and $\beta$ . Over each of the segments $P_iP_{i+1} \ (P_{n+1} = P_1)$ a point $Q_i$ is constructed such that for all $i$ :(i) upon moving from $P_i$ to $P_{i+1}, Q_i$ is seen on the same side of $P_iP_{i+1}$ ,(ii) $\angle P_{i+1}P_iQ_i = \alpha,$ (iii) $\angle P_iP_{i+1}Q_i = \beta.$ Furthermore, let $g$ be a line in the same plane with the property that all the points $P_i,Q_i$ lie on the same side of $g$ . Prove that $$\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g).$$ where $d(M,g)$ denotes the distance from point $M$ to line $g.$