10.109. On the sides of an equilateral triangle, squares are constructed outside it. Their vertices, lying outside the triangle, are sequentially connected. Determine the area of the resulting hexagon if the side of the given triangle is $a$.
Solution.

The area of the desired hexagon (Fig. 10.105)

$S_{H K D E F G}=S_{\triangle A B C}+3 S_{A H K B}+3 S_{\triangle K B D}=\frac{a^{2} \sqrt{3}}{4}+3 a^{2}+3 \cdot \frac{1}{2} a^{2} \sin 120^{\circ}=$ $=\frac{a^{2} \sqrt{3}}{4}+3 a^{2}+\frac{3}{2} \frac{a^{2} \sqrt{3}}{2}=a^{2}(3+\sqrt{3})$.

Answer: $\quad a^{2}(3+\sqrt{3})$.