  The Cayley-Menger variety is the Zariski closure of the set of vectors
specifying the pairwise squared distances between $n$ points in $\mathbb{R}^d$.
This variety is fundamental to algebraic approaches in rigidity theory. We
study the tropicalization of the Cayley-Menger variety. In particular, when $d
= 2$, we show that it is the Minkowski sum of the set of ultrametrics on $n$
leaves with itself, and we describe its polyhedral structure. We then give a
new, tropical, proof of Laman's theorem.
