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Sum on distances

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October 7, 2010
geometry unsolvedgeometry

Problem Statement

In a plane are given n points Pi (i=1,2,…,n)P_i \ (i = 1, 2, \ldots , n) and two angles α\alpha and β\beta. Over each of the segments PiPi+1 (Pn+1=P1)P_iP_{i+1} \ (P_{n+1} = P_1) a point QiQ_i is constructed such that for all ii: (i) upon moving from PiP_i to Pi+1,QiP_{i+1}, Q_i is seen on the same side of PiPi+1P_iP_{i+1}, (ii) ∠Pi+1PiQi=α,\angle P_{i+1}P_iQ_i = \alpha, (iii) ∠PiPi+1Qi=β.\angle P_iP_{i+1}Q_i = \beta. Furthermore, let gg be a line in the same plane with the property that all the points Pi,QiP_i,Q_i lie on the same side of gg. Prove that ∑i=1nd(Pi,g)=∑i=1nd(Qi,g).\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g). where d(M,g)d(M,g) denotes the distance from point MM to line g.g.
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