voronoi, delaunay and weaverbird dual from given set of points

got quite an abstract wish concerning delaunay-triangulation and voronoi-diagramms.

i have a given set of poits, which is fixed and cannot be changed (in the attached definition represented as a popgeo of a planesrf). these points should be connected in a special way, giving every point 3 lines spreading from it.

1) delaunay-edges
if i connect these points with delaunay edges, the count of lines starting from each point is:
a) irregular
b) always higher than 3

2) weaverbird dual graph
if generating the dual graph of this delaunay mesh using the "weaverbird dual graph", i get a voronoi diagramm, which would be perfect for my needs ... if only ... yes i know, its against voronoi definition ... if only the voronoi edges would connect the original points instead of using it as "cell centers".
as the borders of the voronoi cells are defined as "the set of all points with the same distance to two of its center points (my original points)", i guess its impossible to find what i am looking for as a voronoi.
but as i am no mathematician, there is still some hope, that i am simply missing something. the "mesh" i am looking for does not necessarily need to be a voronoi. my wishes are:
a) the numbers of "edges" starting from my points should be 3
b) the largest angle between two of these 3 "edges" should not exceed 180°(even better:175°)

any help or enlightement on this topic is very much appreciated!
roberto

BTW (also in the attached gh sample):
when using the grasshopper "voronoi diagramm" component (applied to my original points), the result is different from  the "weaverbird dual graph" (applied to a delaunay of my original points, as explained above). according to the wikipedia-definition ("the voronoi diagram of a set of points is dual to its delaunay triangulation"), i expected it would be the same. in fact the weaverbird output is much more convenient for my needs. can someone explain the difference in how the two components work?