Central configurations play a basic role in the studies of the n-body problem.
In case of a central configuration, the resulting force on each body always directs towards
the center of mass of the system, and the bodies perform motions in self-similar configurations.
The central configurations of the three-body problem are the Eulerian-Lagrangian solutions.
For n>3 the problem is unsolved, and only particular solutions have been obtained. However,
recently we could derive an analytical solution for an axial-symmetric case of the four-body problem,
giving all solutions in this case. The talk describes the way leading to this analytical solution, reveals
the wonderful world of the kite central configurations and their connections with the Lagrangian
solutions.
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