© 2024 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the ''https://creativecommons.org/licenses/by/4.0/''Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP3.Several continuum and lattice investigations of the QCD three-gluon vertex have recently exposed its key properties, some intimately connected with the low-momentum behavior of the two-point gluon Green's function and especially relevant for the emergence of a mass scale in this latter, via the Schwinger mechanism. In the present study, we report on a lattice determination of the Landau-gauge, transversely projected three-gluon vertex, particularly scrutinizing an outstanding one of these properties, termed planar degeneracy, exploring its implications and capitalizing on it to gain further insight on the low-momentum running of both the three-gluon vertex and its associated strong coupling.