Both theorems are proved in the appendix. Theorem 2 is similar to a result given by Brodal et al. , and is easy to prove. The basic idea is that for any two non-sister leaves {a,b}, there is a pair of anchors such that in the resulting quartet, a and b are not sisters, and the quartet length is exactly the same as the distance between the two leaves minus their terminal branches. We note that similar to us, Brodal et al. use the concept of anchors, but instead of using anchors to define distances, they use anchors to efficiently build Buneman trees from given distances. Thus, despite some parallels, our anchoring mechanism is novel; In particular, Brodal et al. do not prove our surprising result that a single arbitrarily chosen pair of anchors gives additive distances for the correct topology.