$ (A). (B), (C)$ are three spheres in that their centers are not collinear. $ n\le 8$ is the number of planes that touch three spheres. $ A_i,B_i,C_i$ is the point that plane $ \pi_i;i = \overline{1,n}$ touch the spheres $ (A),(B),(C)$. Let $ O_i$ be circumcenter of $ \triangle{A_iB_iC_i};i = \overline{1,n}$. Prove that $ O_i;i = \overline{1,n}$ are collinear. SOLUTION: Denote $ \rho_x(P)$ the power of point $ P$ with respect to sphere $ (X); X\in\{A;B;C\}$. If $ O_i$ circumcenter of $ \triangle{A_iB_iC_i};i = \overline{1,n}\Rightarrow |O_iA_i| = |O_iB_i| = |O_iC_i|\Rightarrow$ $ \Rightarrow \rho_a(O_i) = \rho_b(O_i) = \rho_c(O_i);i = \overline{1,n}\Rightarrow O_i\in d$aopsnowrap][math=inline]$ d$[/math]=[/aopsnowrap]radical axis of three spheres [math=inline]$ (A), (B), (C)$[/math.