Why Euclidean space isn't (Gromov) Hyperbolic?

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I'm trying to understand why the hyperbolic plane is Gromov Hyperbolic while the Euclidean plane isn't. I guess that there is connection to the fact that every similar triangles in $\mathbb{H}^2$ are congruence triangles.

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Hint: What's the thinness constant for an equilateral triangle of side length 1,000,000,000,000,000?