On lattice triangles satisfying $\boldsymbol{B(T)=3}$ with collinear interior lattice points

A lattice point in $\mathbb{R}^2$ is a point $(x,y)$ with $x,y\in\mathbb{Z}$, and a lattice triangle is a triangle whose three vertices are all lattice points. We investigate the integers $k$ with the property that if $T$ is a lattice triangle with $3$ boundary points and $k$ points in the interior, then all $k$ boundary points must be collinear.