Non-Midpoint Segment Splitting in Ruppert's Delaunay Triangulation Refinement Algorithm
Alper Ungor did some nice work related to this issues of how do Delaunay refinement with alternative points. See his paper: 

A. Ungor. Off-centers: A new type of Steiner points for computing size-optimal guaranteed-quality Delaunay triangulations. Computational
Geometry: Theory and Applications (CGTA), 42(2): 109--118, 2009.