  The triangulations of a surface $\Sigma$ with a prescribed set of vertices
can be endowed with a graph structure $\mathcal{F}(\Sigma)$. Its edges connect
two triangulations that differ by a single arc. It is known that, when $\Sigma$
is a convex polygon or a topological surface, the subgraph
$\mathcal{F}_\varepsilon(\Sigma)$ induced in $\mathcal{F}(\Sigma)$ by the
triangulations that contain a given arc $\varepsilon$ is strongly convex in the
sense that all the geodesic paths between two such triangulations remain in
that subgraph. Here, we provide a related result that involves a triangle
instead of an arc, in the case when $\Sigma$ is a convex polygon. We show that,
when the three edges of a triangle $\tau$ appear in (possibly distinct)
triangulations along a geodesic path, $\tau$ must belong to a triangulation in
that path. More generally, we prove that certain $3$-dimensional triangulations
related to the geodesics in $\mathcal{F}(\Sigma)$ are flag when $\Sigma$ is a
convex polygon with flat vertices, and provide two consequences. The first is
that $\mathcal{F}_\varepsilon(\Sigma)$ is not always strongly convex when
$\Sigma$ is a convex polygon with either two flat vertices or two punctures.
The second is that the number of arc crossings between two triangulations of a
topological surface $\Sigma$ does not allow to approximate their distance in
$\mathcal{F}(\Sigma)$ by a factor of less than $3/2$.
