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Three polygonal discs

Source: Kürschak 2013, problem 2

July 6, 2014
geometrygeometry unsolved

Problem Statement

Consider the closed polygonal discs P1P_1, P2P_2, P3P_3 with the property that for any three points A∈P1A\in P_1, B∈P2B\in P_2, C∈P3C\in P_3, we have [△ABC]≤1[\triangle ABC]\le 1. (Here [X][X] denotes the area of polygon XX.) (a) Prove that min⁡{[P1],[P2],[P3]}<4\min\{[P_1],[P_2],[P_3]\}<4. (b) Give an example of polygons P1,P2,P3P_1,P_2,P_3 with the above property such that [P1]>4[P_1]>4 and [P2]>4[P_2]>4.
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