Theorem 1 states anchored distances induced from an additive matrix will correspond to the same topology as the initial matrix (albeit with wrong branch lengths). This result is surprising, but its usefulness might be less clear. Theorem 1 enables new estimators of the tree topology that rely on quartets to compute pairwise distances. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D denote the input data to be used for inferring a phylogeny. Regardless of the nature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D, we require having a quartet estimator. A quartet estimator is a function that given a quartet of leaves Q, uses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D to estimate the quartet tree topology and the quartet length τ(Q), and is statistically consistent if, as the size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D increases, the estimated quartet topology and length both converge in probability to correct values. Statistically consistent quartet estimators can be designed for various models (e.g., sequence evolution  and the MSC [33, 34]).