I have a doubt on Freudenthal Triangulation. I want to partition a simplex into finer simplices.

The FT gives me the vertices of the simplices which partition my original simplex into finer simplices.

I am interested in enumerating over each of the simplices obtained from the FT algorithm.

However, I am not able to conclude that given the vertices, how do I construct the dividing simplices out of them. In other words, if the vertices are given to me then how can I find which of these vertices make one of the simplices (that partitioned the original simplex)

For example consider the case of three dimensions. A simplex in 3-D is given as a convex hull of the vertices (0,0,1), (0 1 0) and (1,0,0). An exemplary FT of this example is the simplices with vertices v1 = (0,0,1), v2 = (0,1,0), v3 = (1,0,0), v4 = (0.5,0.5,0), v5 = (0,0.5,0.5) and v6 = (0.5,0,0.5).

Not sure how to draw it here but on paper you will see the original simplex (which will be a triangle) divided into 4 sub-triangles with the vertices (v1,v5,v6), (v2,v4,v5),(v3,v4,v6) and (v4,v5,v6)

My concern is that given the vertices v1,..,v6, how do I arrive at those 4 sub-simplices. I need it as I will be dealing with higher dimensions and finding the sub-simplices may not be that obvious in those cases.

I believe it might be a simple question with simple solution but I am not able to figure it out. Can you please point me to some reference. Thank You