  Since the strong degeneracies present in the N-body problem, even in the
basic case of the planar three-body problem, nobody inspects the problem of
nonlinear stability of Lagrange relative equilibrium. We introduce a new
coordinate system to reduce degeneracies according to intrinsic symmetrical
characteristic of the N-body problem, then we prove that Lagrange relative
equilibrium is stable in the sense of measure, provided it is spectrally stable
and except six special resonant cases. Indeed, under this condition, there are
abundant KAM invariant tori or quasi-periodic solutions near Lagrange relative
equilibrium. Furthermore, these tori or quasi-periodic solutions form a set
whose relative measure rapidly tends to 1. We also prove that Lagrange relative
equilibrium is exponentially stable for almost every choice of masses in the
sense of measure, provided it is spectrally stable; and topologically, this is
also right for a large open subset of spectrally stable space of masses.
