We credit Lemma 3 in this paper to Fekete and Meijer (2000).  In fact,
it is much older.  Similar constructions appear in 

R. Buck and E. Buck. "Equipartitioning of Convex Sets". 
Math. Mag. 22 (4), 195-198 (1949).

R. Courant and H. Robbins. What is Mathematics?
Oxford University Press, (1941).