Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that 

1) they all share the common vertex M 

2) the simplices $\Delta_i$ triangulate the polytope $P=\mathrm{Conv}(\{V_{i,j}\}_{i=1,…,m|j=1,…,d}) $containing M in its interior?

If the answer is positive, is there a simple algorithm to produce an instance of the points V
given m and d?