In this and the next chapters, we will introduce in more detail the ideas and methods used to solve the Aleksandrov–Rassias problems for each case. Section 5.1 focuses on examining the Aleksandrov–Rassias problem, where the relevant mapping preserves two distances, 1 and \(\sqrt {3}\) . This case is a special case among cases where the ratio of two distances is not an integer. In Sect. 5.2, we consider in detail the Aleksandrov–Rassias problem where the relevant mapping preserves the distances 1 and \(\sqrt {2}\) . Section 5.3 investigates the conditions under which the relevant mapping preserving three distances necessarily become an isometry. As shown in Problem 4.2 \((ii)\) , this problem is also closely related to the Aleksandrov–Rassias problems. In this chapter, we extract the main results from the papers by Rassias and Xiang (Univ Beograd Publ Elektrotehn Fak 11(4):1–8, 2000), Xiang (Aleksandrov problem and mappings which preserve distances, in Functional Equations and Inequalities, ed. by Th.M. Rassias, pp. 297–323, Kluwer, Alphen aan den Rijn, 2000), Xiang (J Math Anal Appl 254(1):262–274, 2001) and organize them so that readers can easily understand them.