  We present novel results for the three-gluon vertex, obtained from an
extensive quenched lattice simulation in the Landau gauge. The simulation
evaluates the transversely projected vertex, spanned on a special tensorial
basis, whose form factors are naturally parametrized in terms of individually
Bose-symmetric variables. Quite interestingly, when evaluated in these
kinematics, the corresponding form factors depend almost exclusively on a
single kinematic variable, formed by the sum of the squares of the three
incoming four-momenta, $q$, $r$, and $p$. Thus, all configurations lying on a
given plane in the coordinate system $(q^2, r^2, p^2)$ share, to a high degree
of accuracy, the same form factors, a property that we denominate \emph{planar
degeneracy}. We have confirmed the validity of this property through an
exhaustive study of the set of configurations satisfying the condition $q^2 =
r^2$, within the range $[0, 5\, \rm GeV]$. Moreover, a preliminary exploration
reveals that the planar degeneracy persist in the case of more arbitrary
configurations. This drastic simplification allows for a remarkably compact
description of the main bulk of the data, which is particularly suitable for
future numerical applications. A semi-perturbative analysis reproduces the
lattice findings rather accurately, once the inclusion of a gluon mass has
cured all spurious divergences.
