HMMT 2010 E9

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Three unit circles $\omega_{1}, \omega_{2}$ , and $\omega_{3}$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds the three circles in such a way that each of its four sides is tangent to at least one of $\omega_{1}, \omega_{2}$ and $\omega_{3}$ . Find the side length of the square $S$ .

Dataset text NuminaMath olympiads. Quoted with source.

Also HMMT 2010 G6.

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\frac{\sqrt{6}+\sqrt{2}+8}{4}

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