In 1970, A. D. Aleksandrov asked whether a mapping must be an isometry if it preserves a certain distance. As we saw in the previous chapter, F. S. Beckman and D. A. Quarles solved this problem for mappings from an n-dimensional Euclidean space into the same one. After weakening the result of the Beckman–Quarles theorem to make the proof easier, E. M. Schröder proved in 1979 that any mapping that preserves two distances \(\rho \) and \(2\rho \) is an affine isometry. In a situation where the Beckman–Quarles theorem is already known, Schröder’s theorem by itself is of little significance. Schröder, however, presented a novel idea like the m-chain in the process of proving his theorem, and using this idea, W. Benz was able to significantly expand the Beckman–Quarles theorem. In this chapter, we present in detail three historically important theorems, Schröder’s theorem, Benz’s theorem, and Benz–Berens theorem.