KöMaL 2017 A702

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?