1.16. Each of the three equal circles of radius $r$ touches the other two. Find the area of the triangle formed by the common external tangents to these circles.
1.16. Triangle $ABC$ is equilateral (Fig. 1.14); therefore, its area is $\frac{a^{2} \sqrt{3}}{4}$, where $a$ is the side of the triangle. Draw $OK \perp AC$. Since $\angle OAK=30^{\circ}$, then $AO=2r$, $AK=r \sqrt{3}$, hence $a=2r \sqrt{3} + 2r = 2r(\sqrt{3} + 1)$. Therefore,

$$
S=\frac{4r^{2}(3+2\sqrt{3}+1)\sqrt{3}}{4}=2r^{2}(2\sqrt{3}+3)
$$

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

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-421.jpg?height=404&width=462&top_left_y=1165&top_left_x=46)

Fig. 1.15