KöMaL 2020 B5131

Let $H$ be an equilateral triangle of unit area, let $O$ be a fixed point, and for any point $P$ let $H_P$ denote the triangle obtained from triangle $H$ by a parallel shift with vector $\overrightarrow{OP}$. Consider the set $N$ of points $P$ for which the area of the intersection $H\cap H_P$ is at least $4/9$. What is the area of $N$?