## Co.combinatorics – Existence of a "sufficiently generic" lattice-point interior to a lattice triangle

Leave $$T$$ be a lattice triangle in $$Bbb R ^ 2$$. Is it always possible to find a point? $$p in T cap Bbb Z ^ 2$$ fulfilling the following condition:

(*) $$operatorname {conv} (p, v) cap Bbb Z ^ 2 = {p, v }$$ for each vertex $$v in T$$, where $$operatorname {conv}$$ It's the convex hull.

For example: In this image:

$$p = D$$ It works, while in this photo:

$$p = D$$ It does not work because the line segment $$DC$$ cross $$E$$.