The maximum separation problem is to find the maximum of the minimum pairwise distance of n points in a planar body \({\mathcal {B}}\) on the Euclidean plane. In this paper this problem will be considered if \({\mathcal {B}}\) is the equilateral triangle of side length 1 and the number of points is 13. We will present the exact separation distance of 13 points in the equilateral triangle of side length 1 and we will prove a conjecture of Melissen from 1993 and a conjecture of Graham and Lubachevsky from 1995.

中文翻译：

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将13个圆排成一个等边三角形

最大分离问题是在欧几里得平面上找到平面物体\（{\ mathcal {B}} \）中n个点的成对最小距离中的最大值。在本文中，如果\（{\ mathcal {B}} \）是边长为1的等边三角形且点数为13，则将考虑此问题。我们将在等边三角形中给出13个点的精确分隔距离边长为1的情况，我们将证明1993年的梅利森猜想和1995年的Graham和Lubachevsky猜想。