RU-ALLRUSSIAN 1983 Problem 3

$\mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3}$ are circles, none of which lie inside either of the others. $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ touch at Z, $\mathrm{C}_{2}$ and $\mathrm{C}_{3}$ touch at X, and $\mathrm{C}_{3}$ and $\mathrm{C}_{1}$ touch at Y. Prove that if the radius of each circle is increased by a factor $2 / \sqrt{3}$ without moving their centers, then the enlarged circles cover the triangle XYZ.