$n$-gons on grid with $1$ internal grid point

Question

Investigating the triangle with $1$ internal point:

There are at least $5$ triangles:

EDITED: According to comment adding the $3$-rd case:

Is it known how many triangles and $n$-gons in general exist with $1$ internal point.

Answer 1

I think 3 is the last interesting $n$.

For $n=4$, consider the (possibly concave) quadrilateral $(0,0), (2,0), (1,2), (0,y)$ for arbitrary positive $y\ne 4$, which encloses only $(1,1)$.

Or this convex quadrilateral $(1,0), (x, 1), (-1,0), (-x, -1)$ for arbitrary $x$, which encloses only $(0,0)$.