I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps MO would be a more suitable place to attract answers.

Let $T$ be an equilateral triangle of unit area, with vertices $A_1, A_2, A_3$. Place triangles $T_1, T_2, T_3$ each of unit area such that the centroid $G_i$ of $T_i$ is equal to $A_i$ for $i = 1,2,3$. What is the maximum possible value of the area of the region $T_1 \cap T_2 \cap T_3$, and what configuration and shape of $T_1, T_2, T_3$ achieves this maximum?