  Triple factorisations of finite groups $G$ of the form $G=PQP$ are essential
in the study of Lie theory as well as in geometry. Geometrically, each triple
factorisation $G=PQP$ corresponds to a $G$-flag transitive point/line geometry
such that `each pair of points is incident with at least one line'. We call
such a geometry \emph{collinearly complete}, and duality (interchanging the
roles of points and lines) gives rise to the notion of \emph{concurrently
complete} geometries. In this paper, we study triple factorisations of the
general linear group $\mathrm{GL}(V)$ as $PQP$ where the subgroups $P$ and $Q$
either fix a subspace or fix a decomposition of $V$ as $V_1\oplus V_2$ with
$\dim(V_{1})=\dim(V_{2})$.
