collapsed by $(\varphi_n)$ directly comes from the observation that the two sequences $\left(\varphi_n \operatorname{\mathrm{proj}}_{c'_n}(C_n)\right) = \left(\operatorname{\mathrm{proj}}_{\varphi_n c'_n}(\varphi_n C_n)\right)$ and $\left(\varphi_n \operatorname{\mathrm{proj}}_{c'_n}(D_n)\right) = \left(\operatorname{\mathrm{proj}}_{\varphi_n c'_n}(\varphi_n D_n)\right)$ have the same limit by Proposition \[proposition:projections\].

We draw attention to the fact that it is essential to have $\tilde{c}$ and $\tilde{c}'$ opposite each other to apply the previous lemma.

\[proposition:centerok\] Let $\Delta$ be a thick compact polygon and $A_n \to A$ be a converging sequence of apartments of $\Delta$. If there exist two sequences $a_n \to a$ and $b_n \to b$ in $\operatorname{\mathrm{Vert}}\Delta$ collapsed by a sequence $\{\varphi_n\} \subseteq \operatorname{\mathrm{Auttop}}(\Delta)$ and if $\varphi_n A_n \to \tilde{A}$ for some apartment $\tilde{A}$ of $\Delta$, then there exist two sequences $a'_n \to a'$ and $b'_n \to b'$ centered at $c'_n \to c'$ and collapsed by some subsequence of $(\varphi_n)$, with $c'_n \in A_n$ for all $n \in \operatorname{\mathbf{N}}$.

By Proposition \[proposition:centered\], we can assume that the two sequences $a_n \to a$ and $b_n \to b$ are centered at some sequence of vertices $c_n \to c$. Let $d$ be a vertex of $A$ opposite $c$ (to find such a vertex, one can take a gallery from $c$ to any vertex of $A$ and extend this one with chambers 