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Sizing up Equilateral Shapes

Source: Cono Sur 2023 #3

August 7, 2023
geometrygeometric inequalitycono sur

Problem Statement

In a half-plane, bounded by a line rr, equilateral triangles S1,S2,…,SnS_1, S_2, \ldots, S_n are placed, each with one side parallel to rr, and their opposite vertex is the point of the triangle farthest from rr.
For each triangle SiS_i, let TiT_i be its medial triangle. Let SS be the region covered by triangles S1,S2,…,SnS_1, S_2, \ldots, S_n, and let TT be the region covered by triangles T1,T2,…,TnT_1, T_2, \ldots, T_n.
Prove that area(S)≤4⋅area(T).\text{area}(S) \leq 4 \cdot \text{area}(T).
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