We show the existence of central configurations in the planar five-body problem where four bodies are located at the vertices of a rhombus, called rhombus plus one central configurations. Concretely we prove analytically their existence when one diagonal is nearly equal to the sides of the rhombus and when the two diagonals are either equal or nearly equal. In addition, we prove that given a rhombus plus one configuration, the corresponding vector of positive masses that makes the configuration central, if exists, is unique.

中文翻译：

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五体问题的中心配置：Rhombus Plus One

我们显示了平面五体问题中中心构型的存在，其中四个体位于菱形的顶点（称为菱形加一个中心构型）上。具体来说，当一个对角线几乎等于菱形的边且两个对角线相等或几乎相等时，我们通过分析证明了它们的存在。另外，我们证明给定菱形加一个构型，使构型居中的正质量矢量（如果存在）是唯一的。