MathDB

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

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.

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