Estonian Mathematical Olympiad

Each interior angle of a regular hexagon has a size of $120^\circ$. Thus, a regular hexagon can be divided into $6$ equilateral triangles with side lengths equal to the hexagon itself (Fig. 23). Denoting the area of such a triangle as $S$, the area of the hexagon $ABCDEF$ is therefore $6S$.

The opposite sides of a rectangle are of equal length. Hence $XY = EF$, $YZ = AB$ and $XZ = CD$, so $XYZ$ is also an equilateral triangle with the same side length. Since the rectangles $ABZY$, $CDXZ$ and $EFYX$ are equal, the triangles $AYF$, $BZC$ and $DXE$ are isosceles. Their bases are the sides of the hexagon, the base angle is $120^\circ - 90^\circ = 30^\circ$ and the vertex angle is $180^\circ - 2 \cdot 30^\circ = 120^\circ$. Thus, it is possible to form one equilateral triangle from these three triangles, whose side length is equal to the side length of the hexagon (Fig. 24).

In total, the area not covered by the rectangles is $2S$. Therefore, the total area of the rectangles is $6S - 2S = 4S$, which is $\frac{2}{3}$ of the area of the hexagon.

Fig. 23

Fig. 24