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cos \phi >= 1-sin(a/2), constructing triangles by triangles

Source: III Soros Olympiad 1996-97 R3 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 31, 2024
geometrycombinatorial geometrygeometric inequalitytrigonometry

Problem Statement

There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than aa, and ϕ\phi, where ϕ\phi is the largest angle of the resulting triangle, then cos⁡ϕ≥1−sin⁡(a/2)\cos \phi \ge 1-\sin (a/2).
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