We study five-body central configurations. The three-body equilateral triangle Lagrange’s central configuration and four-body equilateral tetrahedron Lehmann-Filhès’s central configuration were generalized to an equilateral five-body central configuration in four dimensions. Our working tool is a coordinate system that had many useful properties in considering central configurations of three, four, and five bodies. From this equilateral solution, projecting to three dimensions, we compute concave and convex five-body central configurations of five different positive masses. We discover the possibility of geometric symmetry of these solutions when two of the masses are equal. Then we study the plane symmetric central configurations with a kite figure. This figure has three masses on the symmetry axis of the kite and two equal masses symmetrically placed on a transversal orthogonal line to the axis. Our parameters are now the masses and the distances along the axis of symmetry. The distances are measured with respect to the crossing point of the diagonals. In this paper, we find and solve a linear equation for these three masses in terms of the distances. We will draw an example of typical elliptic trajectories obeying the Newton equations of motion in such a kite central configuration.

中文翻译：

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五体中央结构和对称性

我们研究五体中央结构。将三体等边三角形Lagrange的中心构型和四体等边四面体Lehmann-Filhès的中心构型在四个维度上推广为等边五体中心构型。我们的工作工具是一个坐标系统，在考虑三个，四个和五个实体的中心配置时，具有许多有用的属性。从这个等边解，投影到三个维度，我们计算出五个不同正质量的凹凸五体中心构型。当两个质量相等时，我们发现了这些解的几何对称性的可能性。然后，我们用风筝造型研究平面对称的中心配置。该图在风筝的对称轴上具有三个质量，并且在与该轴垂直的横向正交线上对称放置两个相等的质量。现在，我们的参数是质量和沿对称轴的距离。相对于对角线的交叉点测量距离。在本文中，我们根据距离找到并求解了这三个质量的线性方程。我们将在风筝中心配置下遵循牛顿运动方程画出一个典型的椭圆轨迹示例。