$n$-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.

What about the triangle $(0,0), (1,0), (2, 4)$? $(0,0), (3,0), (1,2)$? — peterwhy, Jul 26 '17 at 18:09
Is this infinite? Just trying to find some literature on this. — Gevorg Hmayakyan, Jul 26 '17 at 18:10
You can draw a line from the internal point to each of the 3 outside points .... there - you have three triangles. But I don't quite understand the question (especially about the n-gons) — Josh, Jul 26 '17 at 18:13
Dear Josh the question is how many triangles with vertices on grid exist with 1 internal grid point. Is this finite? If yes the exact count. — Gevorg Hmayakyan, Jul 26 '17 at 18:16

peterwhy · Accepted Answer · 2017-07-27T13:36:24.830

3

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 $(0,0)$.

edited Jul 27 '17 at 13:36

answered Jul 26 '17 at 18:32

peterwhy

1

Maybe one should add the condition that the polygon is convex in order to make the cases $n\geq4$ interesting again. – Christian Blatter Jul 26 '17 at 19:27
Yes this proves that for $n=4$ the solutions are infinite. Obviously the same is for $n>=5$. I think the next question is 2 internal point case. – Gevorg Hmayakyan Jul 26 '17 at 22:32
And also are the cases listed for triangle cover all cases? – Gevorg Hmayakyan Jul 26 '17 at 22:32
1

@ChristianBlatter I have added a family of convex quadrilaterals that enclose only one lattice point. – peterwhy Jul 27 '17 at 13:37
Amazingly nice! – Gevorg Hmayakyan Jul 27 '17 at 19:11

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