be a thick compact polygon. If there exist two sequences $a_n \to a$ and $b_n \to b$ in $\operatorname{\mathrm{Vert}}\Delta$ collapsed by a sequence $\{\varphi_n\} \subseteq \operatorname{\mathrm{Auttop}}(\Delta)$, then there exist two sequences $a'_n \to a'$ and $b'_n \to b'$ collapsed by $(\varphi_n)$ and such that $\operatorname{\mathrm{D}}(a'_n, b'_n) = \operatorname{\mathrm{D}}(a', b') = 2$ for all $n \in \operatorname{\mathbf{N}}$.

See [@Burns]\*[Discussion between Assertions 2.3 and 2.4]{}.

Now suppose that we are given a converging sequence of apartments $A_n \to A$, i.e. $2m$ converging sequences $v_i^{(n)} \to v_i$ ($i \in \{0, \ldots, 2m-1\}$) where $v_0^{(n)}, \ldots, v_{2m-1}^{(n)}$ (resp. $v_0, \ldots, v_{2m-1}$) are the vertices of an apartment $A_n$ (resp. $A$). In this context, we can generally even assume that the middle vertex $c_n$ of $a_n$ and $b_n$ (which are such that $\operatorname{\mathrm{D}}(a_n, b_n) = 2$) is a vertex of $A_n$. For this reason we introduce the following definition.

Let $\Delta$ be a compact polygon. Two convergent sequences $a_n \to a$ and $b_n \to b$ in $\operatorname{\mathrm{Vert}}\Delta$ are said to be **centered at** $c_n \to c$ if $\operatorname{\mathrm{D}}(a_n, c_n) = \operatorname{\mathrm{D}}(c_n, b_n) = 1$ for all $n \in \operatorname{\mathbf{N}}$. This also implies that $\operatorname{\mathrm{D}}(a, c) = \operatorname{\mathrm{D}}(c, b) = 1$.

Before proving the announced result, 