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Area of triangle >= SQRT(2)

Source: IMO Shortlist 1989, Problem 28, ILL 87

September 18, 2008
geometryarea of a trianglegeometric inequalityIMO Shortlistpoint set

Problem Statement

Consider in a plane P P the points O,A1,A2,A3,A4 O,A_1,A_2,A_3,A_4 such that \sigma(OA_iA_j) \geq 1   \forall i, j \equal{} 1, 2, 3, 4, i \neq j. where σ(OAiAj) \sigma(OA_iA_j) is the area of triangle OAiAj. OA_iA_j. Prove that there exists at least one pair i0,j0∈{1,2,3,4} i_0, j_0 \in \{1, 2, 3, 4\} such that σ(OAiAj)≥2. \sigma(OA_iA_j) \geq \sqrt{2}.
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