Comment by rosshemsley

Perhaps an easier (and more robust) way to do this is as follows:

1) Take your points in 3D, and add the point at the origin

2) Compute Delaunay in 3D

3) The 2D faces not containing the origin is your triangulation.

To get the Voronoi tessellation:

1) Circulate the faces of each vertex in order (use orientation test if they are not ordered yet)

2) Connect the circumcenters of each 2D face you visit.

Voilà!