We present a novel parameter-free cosmological void finder (DIVE, Delaunay TrIangulation Void findEr) based on Delaunay Triangulation (DT), which efficiently computes the empty spheres constrained by a discrete set of tracers. We define the spheres as DT voids, and describe their properties. We apply this technique on 100 PATCHY mock halo catalogues with volumes of 2.5 h−1Gpc side each, with a bias and number density similar to the BOSS CMASS Luminous Red Galaxies. Our results show that there are two main species of DT voids, which can be characterised by the radius: they have different responses to halo redshift space distortions, to number density of tracers, and reside in different dark matter environments. The power spectra of DT voids show that the bias of the two populations are different, demonstrating that the small DT voids are essentially tracers of groups of haloes. We further resolve DT voids from the largest recently publicly available sample of Luminous Red Galaxies from SDSS-III BOSS DR11, and unveil for the first time a >3σ BAO detection from large voids in observations.