3 interior points in a grid based polygon

Question

Given a polygon with vertices on a grid with 3 interior grid points and no 3 vertices lying on the same line.

Is it true that all vertices are on the same circle?

EDITED

There is also another counter example with convex points:

Answer 1

The answer is no.

Consider the figure that you provided, remove the point $H$ from the polygon and move the point $J$ of the polygon one space up to become $J'$, so that it is the grid point adjacent to the points $I$ and $J$. Than the polygon $J'IGLKJ'$ contains the only three internal grid points $M,N,O$ and the vertices clearly do not lie on a circle.

Answer 2

Would this qualify as a counterexample (J - I - H - G - L and then the red sides)? It is a polygon, it has vertices on a grid, 3 interior points, and no 3 vertices lying on the same line.