Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.