We study the geometry of convex lattice $n$-gons with $n$ boundary lattice points and $k\geq 3$ collinear interior lattice points. We describe a process to construct a primitive lattice triangle from an edge of a convex lattice $n$-gon, hence adding one edge in a way so that the number of boundary points increases by $1$, while the number of interior points remains unchanged. We also present the necessary conditions to construct such a primitive lattice triangle, as well as an upper bound for the number of times this is possible. Finally, we apply the previous results to fully classify the positive integers for which there exists a convex $n$-gon with $k$ collinear and non-collinear interior points.