IMO 1983 LL GDR34

Origin: GDR

Problem

In a plane are given n points Pi (i = 1, 2, . . . , n) and two angles \alpha and \beta. Over each of the segments PiPi=1 (Pn+1 = P1) a point Qi is constructed such that for all i: (i) upon moving from Pi to Pi+1, Qi is seen on the same side of PiPi+1, (ii) \anglePi+1PiQi = \alpha, (iii) \anglePiPi+1Qi = \beta. Furthermore, let g be a line in the same plane with the property that all the points Pi, Qi lie on the same side of g. Prove that n  i=1 d(Pi, g) = n  i=1 d(Qi, g), where d(M, g) denotes the distance from point M to line g.