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4 points, 6 lines, prove or disprove that the segments form a hexagon

Source: Austrian-Polish 1983

May 16, 2019
combinatorial geometrygeometryhexagon

Problem Statement

Let P1,P2,P3,P4P_1,P_2,P_3,P_4 be four distinct points in the plane. Suppose ℓ1,ℓ2,…,ℓ6\ell_1,\ell_2, … , \ell_6 are closed segments in that plane with the following property: Every straight line passing through at least one of the points PiP_i meets the union ℓ1∪ℓ2∪…∪ℓ6\ell_1 \cup \ell_2\cup … \cup\ell_6 in exactly two points. Prove or disprove that the segments ℓi\ell_i necessarily form a hexagon.
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