multiple curves, those curves have to show up in each main geodesic in the same order from either end, therefore we can just take the inner triangle where the edges share no curves and apply Case 3.

**Case 3**: The argument given for this case is similar to the short cut argument in [@MMII]. Assume none of the three main geodesics, $g_{\alpha \beta}, g_{\alpha \delta}$, and $g_{ \beta \delta}$ share a curve. By Theorem \[curve hyp\] there exists a curve $c \in {\mathcal{C}}(S)$ that is distance at most $17$ from $g_{\alpha \beta}, g_{\alpha \gamma}$, and $g_{ \beta \gamma}$; let $c$ be the curve that minimizes the distance from all three main geodesics. Define $v_{\alpha \beta}$ to be the vertex in $g_{\alpha \beta}$ which has the least distance to $c$, and similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$.

Consider the geodesic $[v_{\alpha\beta}, c]_S$ and let $c_0$ be the curve adjacent to $c$ in this geodesic. Let $v_*^{-1}$ be the curve in $g_*$ that precedes $v_{*}$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}\}$ to $\{c, c_0 \}$ with a hierarchy. We denote the main geodesic of this hierarchy as $[c, v_{\beta\gamma}]_S$.

Take a vertex $w \in [c, v_{ \beta \gamma}]_S$ where $w$ is not equal 