KöMaL 2012 B4429

$A_1 B_1 C_1$ and $A_2 B_2 C_2$ are two triangles such that their sides $A_1 B_1$ and $A_2 B_2$, $B_1 C_1$ and $B_2 C_2$, as well as $A_1 C_1$ and $A_2 C_2$ are parallel. Vertex $A_1$ is connected to vertices $B_2$ and $C_2$, vertex $B_1$ is connected to $C_2$ and $A_2$, and vertex $C_1$ is connected to $A_2$ and $B_2$. Given that the areas of the original triangles are $T_1$ and $T_2$, what may be the area of the hexagon formed by the midpoints of the connecting line segments obtained in this way?