we show the next lemma, which provides a way to go from a sequence of vertices to a sequence of opposite vertices.

\[lemma:opposite\] Let $\Delta$ be a compact polygon and let $a_n \to a$ and $b_n \to b$ be two sequences in $\operatorname{\mathrm{Vert}}\Delta$ centered at $c_n \to c$ and collapsed by a sequence $\{\varphi_n\} \subseteq \operatorname{\mathrm{Auttop}}(\Delta)$. Let also $c'_n \to c'$ be a sequence in $\operatorname{\mathrm{Vert}}\Delta$ such that $c'_n$ (resp. $c'$) is opposite $c_n$ (resp. $c$) for all $n \in \operatorname{\mathbf{N}}$. Suppose that $\varphi_n c_n \to \tilde{c}$ and $\varphi_n c'_n \to \tilde{c}'$ for some opposite vertices $\tilde{c}$ and $\tilde{c}'$. Denote by $C_n$ (resp. $D_n$) the chamber whose vertices are $c_n$ and $a_n$ (resp. $b_n$) and by $a'_n$ (resp. $b'_n$) the vertex of $\operatorname{\mathrm{proj}}_{c'_n}(C_n)$ (resp. $\operatorname{\mathrm{proj}}_{c'_n}(D_n)$) different from $c'_n$. Then the sequences $(a'_n)$ and $(b'_n)$ converge, are centered at $(c'_n)$ and are collapsed by $(\varphi_n)$.

![Illustration of Lemma \[lemma:opposite\].[]{data-label="picture:opposite"}](figure2.eps)

By Proposition \[proposition:projections\], the sequence $(a'_n)$ converges to $a'$, the vertex of $\operatorname{\mathrm{proj}}_{c'}(C)$ different from $c'$ where $C$ is the chamber having vertices $c$ and $a$. The sequence $(b'_n)$ converges to $b'$ defined in the same way and $a' \neq b'$ since $a \neq b$. The fact that these two sequences are 