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clockwise sides of convex polygon

Source: Vietnam TST 1999 for the 40th IMO, problem 3

June 26, 2005
vectorgeometryparallelogramrotationgeometry unsolved

Problem Statement

Let a convex polygon HH be given. Show that for every real number a∈(0,1)a \in (0, 1) there exist 6 distinct points on the sides of HH, denoted by A1,A2,…,A6A_1, A_2, \ldots, A_6 clockwise, satisfying the conditions: I. (A1A2)=(A5A4)=a⋅(A6A3)(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3). II. Lines A1A2,A5A4A_1A_2, A_5A_4 are equidistant from A6A3A_6A_3. (By (AB)(AB) we denote vector ABAB)
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