6. Three spheres with a radius of 1 are mutually externally tangent, and each sphere is also externally tangent to two other spheres with a radius of $r$. If these two spheres with a radius of $r$ are also externally tangent to each other, then the value of $r$ is $(\quad)$.
(A) 1
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{1}{6}$
6.D.

Let $O_{1}, O_{2}, O_{3}$ be the centers of three spheres with radius 1, and $I_{1}, I_{2}$ be the centers of two spheres with radius $r$. Then they form a solid figure (as shown in Figure 3), 
$H$ is the centroid of $\triangle O_{1} O_{2} O_{3}$. Since $\triangle O_{1} O_{2} O_{3}$ is an equilateral triangle with side length 2, we have $O_{1} H=\frac{2 \sqrt{3}}{3}$. Also, $\triangle I_{1} O_{1} H$ is a right triangle with $\angle I_{1} H O_{1}$ as the right angle, hence
$$
I_{1} O_{1}^{2}=I_{1} H^{2}+O_{1} H^{2},
$$

which means $(1+r)^{2}=r^{2}+\left(\frac{2 \sqrt{3}}{3}\right)^{2}$
Solving for $r$ gives $r=\frac{1}{6}$.