  A {\it two-dimensional continued fraction expansion} is a map $\mu$ assigning
to every $x \in\mathbb R^2\setminus\mathbb Q^2$
  a sequence $\mu(x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices
$x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in\mathbb Q^2, d_{ni}>0, p_{ni}, q_{ni},
d_{ni}\in \mathbb Z,$ $i=1,2,3$, such that \begin{eqnarray*} \det
\left(\begin{matrix} p_{n1}& q_{n1} &d_{n1}\\ p_{n2}& q_{n2} &d_{n2}\\ p_{n3}&
q_{n3} &d_{n3} \end{matrix} \right) = \pm 1\,\,\, \,\,\,\mbox{and}\,\,\,\,\,\,
\bigcap_n T_n = \{x\}. \end{eqnarray*} We construct a two-dimensional continued
fraction expansion $\mu^*$ such that for densely many (Turing computable)
points $x$ the vertices of the triangles of $\mu(x)$ strongly converge to $x$.
Strong convergence depends on the value of $\lim_{n\to
\infty}\frac{\sum_{i=1}^3\dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$
("dist" denoting euclidean distance) which in turn depends on the smallest
angle of $T_n$. Our proofs combine a classical theorem of Davenport Mahler in
diophantine approximation, with the algorithmic resolution of toric
singularities in the equivalent framework of regular fans and their stellar
operations.
