KöMaL problem A, 2017

Geometry Difficulty 6.0 National Olympiad Find the answer Hungary

Fix a triangle $ABC$ . We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$ , $Y$ lies on segment $CA$ , $Z$ lies on segment $AB$ , and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X$ , $\angle B=\angle Y$ , $\angle C=\angle Z$ ). Of all the elegant triangles, which one has the smallest perimeter?

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