In this chapter, we will generalize the parallelogram law to all cases with a given number of points. In Sect. 6.1, we prove the short diagonals lemma, which is a generalization of the parallelogram law for the given four points in the inner product space. Moreover, applying the short diagonals lemma and the parallelogram law, we prove an inequality for the distances between any two points among the given six points. Section 6.2 is devoted to a generalization of the short diagonals lemma to an inequality for the distances among the given 2n points in the inner product space. In this section, we conceive and prove an inequality involving all distances between an even number of points. It seems to be more difficult to prove an inequality for the distances between any two points among an odd number of points than among an even number of points. In Sect. 6.3, we use a new method other than the short diagonals lemma to prove an inequality for the distances between any two of the given five points. It is somewhat surprising that the golden ratio appears in this inequality. In Sect. 6.4, an inequality is introduced that describes in general the relationship between distances among the given n points. When studying inequalities for distances between given n points, it is recommended to pay attention to the following two requirements: The main results presented in this chapter have been extracted from the papers by Jung (Nonlinear Anal 62(4):675–681, 2005), Jung and Lee (J Math Anal Appl 324(2):1363–1369, 2006), Jung and Nam (J Math Inequal 12(4):1189–1199, 2018), Jung and Nam (J Math Inequal 13(4):969–981, 2019) and explained in detail so that the reader can easily understand them.