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We study the geometry of the space of all non-empty subsets of the real line ℝ of cardinality at most three endowed with the Hausdorff metric. The space is known to be homeomorphic to the 3-dimensional Euclidean space ℝ³. We prove that it is, in fact, (3 + 4π)-bi-Lipschitz equivalent to ℝ³. We also show that all isometries of ℝ³ are induced by isometries of ℝ.