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Elegant triangle with minimal perimeter

Source: KöMaL A. 702

November 11, 2017
geometryinequalitiesoptimizationgeometric inequalitykomal

Problem Statement

Fix a triangle ABCABC. We say that triangle XYZXYZ is elegant if XX lies on segment BCBC, YY lies on segment CACA, ZZ lies on segment ABAB, and XYZXYZ is similar to ABCABC (i.e., ∠A=∠X,∠B=∠Y,∠C=∠Z\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z ). Of all the elegant triangles, which one has the smallest perimeter?
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