Double pyramidal central configurations

For N + 2 (N ≥ 4) body problem, if N bodies are at the vertices of a unit regular polygon base and the (N + 1)th body and the (N + 2)th body have equal masses and are at two ends of the vertical line of the plane P formed by the former N bodies, and the vertical line passes the geometrical center of the unit regular polygon and the geometrical center coincides with the center of masses m₁,..., m_N, and m₁,..., m_N, m_N+1, m_N+2 (that is, the (N + 1)th body and the (N + 2)th body have the same distance from the center of masses), then the N + 2 bodies form a double pyramidal central configuration if and only if the masses which are at the vertices of a regular polygon are equal and (1.6) holds.