I want to divide a n dimensional simplex (convex hull of n+1 points) in n+1 parts. E.g. consider the simplex in 2-D which is the convex hull of points (0,0,1), (0,1,0) amd (1,0,0). It can be divided in three parts as
$P\_1 = \{(0,1,0),(1/2,1/2,0),(0,1/2,1/2),(1/3,1/3,1/3)\}$ 

$P\_2 = \{(1,0,0),(1/2,1/2,,0),(1/2,0,1/2),(1/3,1/3,1/3)\}$

$P\_3 = \{(0,0,1),(1/2,0,1/2),(0,1/2,1/2),(1/3,1/3,1/3)\}$

I can somehow arrive at these partitions by considering intersection of some halfspaces (line joining the vertex to the midpoint of the other side). However, I am unable to extend this to higher dimensions.

Is there an algorithm in literature which does the same partitions. Can someone point me to some reference.

Thanks