# Existence of a “generic enough” lattice point interior to a lattice triangle

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:

$$p=D$$ works, whereas in this picture:

$$p=D$$ does not work because the line segment $$DC$$ passes through $$E$$.

Edit: In light of fedja's counterexample (the triangle with vertices $$(0,0)$$, $$(3,2)$$, and $$(1,3)$$, and at their suggestion, I'm modifying the question to ask if (*) holds when $$T$$ contains "sufficiently many" points in the interior.

• It looks to me like your definition of $T$ is a bit ambiguous, as it first refers to the entire triangle (with interior), but later (possibly) refers to only the three vertices of it. – Marcus M Dec 12 '18 at 21:15
• @MarcusM I'm not sure what you mean. – Avi Steiner Dec 12 '18 at 21:16
• When you say $p \in T \cap \mathbb{Z}^2$, it looks like you mean for $T$ to include the interior of the triangle; however, your condition for $p$ that you seek is more ambiguous. It says $\text{conv}(p,v) = \{p,v\}$ for all $v \in T$, however in your image you only tested in for $v$ being the extreme points (i.e. the three vertices) of the triangle. Is that the condition that you seek? – Marcus M Dec 12 '18 at 21:18
• How about (0,0), (3,2),(1,3) for the vertices of $T$? – fedja Dec 13 '18 at 0:09
• There are general visibility results by Adikhari which may be a start on this. Essentially, if an integer interval is large enough, it contains an integer coprime to a given integer N, and this has implications to your problem. Search for work of Granville in this area too. Gerhard "Will Help Look For Forest" Paseman, 2018.12.12. – Gerhard Paseman Dec 13 '18 at 2:06