  We classify the extensions of n-body central configurations to (n + 1)-body
central configurations in R3, in both the collinear case and the non-collinear
case. We completely solve the two open questions posed by Hampton (Nonlinearity
18: 2299- 2304, 2005). This classification is related with study on co-circular
and co-spherical central configurations. We also obtain a general property of
co-circular central configurations.
