We discuss Lepp-centroid versus Lepp-midpoint algorithms for Delaunay quality triangulation. We present geometrical results that ensure that the centroid version produces triangulations with both average smallest angles greater than those obtained with the midpoint version and with bigger smallest edges, without suffering from a rare looping case associated to the midpoint method. Empirical study shows that the centroid method behaves significantly better than the midpoint version (and than the off-center algorithm for angles bigger than 25 degrees), for geometries whose initial Delaunay triangulation have triangle smallest edges over the boundary. (C) 2008 Elsevier Ltd. All rights reserved.