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:

good

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

enter the description of the image here

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