It is well-known that the side length of a regular hexagon is half the length of its longest diagonals. From this property, one can easily see that for every positive integer $$m>1$$, any regular 6m-gon contains two non-congruent diagonals that are commensurable. In this paper, we show that if n is not a multiple of 6, then all pairs of diagonals of different lengths of a regular n-gon are incommensurable. This yields a characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable. The main result gives positive answers to some questions on this topic.

中文翻译：

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对角线对要么全等要么不可通约的规则 n 边形的表征

众所周知，正六边形的边长是其最长对角线长度的一半。从这个性质可以很容易地看出，对于每一个正整数 $$m>1$$，任何规则的 6m 边形都包含两条可公度的非全等对角线。在本文中，我们证明如果 n 不是 6 的倍数，那么规则 n 边形的所有不同长度的对角线对都是不可公约的。这产生了规则 n 边形的特征，其对角线对要么是全等的，要么是不可通约的。主要结果对有关该主题的一些问题给出了肯定的回答。