Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a point $p\in \operatorname{interior}(T)\cap \Bbb Z^2$ satisfying the following condition: (*) $\operatorname{conv}(p,v)\cap \Bbb Z^2 = \{p,v\}$ for every vertex $v\in T$, where $\operatorname{conv}$ is convex hull. For example: In this picture: [![good][1]][1] $p=D$ works, whereas in this picture: [![enter image description here][2]][2] $p=D$ does not work because the line segment $DC$ passes through $E$. [1]: https://i.sstatic.net/ZIVWh.png [2]: https://i.sstatic.net/0So5d.png