I'm looking for a proof for this problem on simplex which I think it is true
Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center circumscribed sphere is $O.$ Let $O_i$ be the centers circumscribed sphere of the simplex $GA_{0}A_{1}...A_{i-1}A_{i+1}...A_n$. Prove that $O$ is centroid of the simplex $O_0O_1...O_n.$
Let $O$ be the origin, $\vec{OA_i}=a_i$, $\vec{OO_i}=p_i$, $\vec{OG}=a$.
Then $\|a_i\|=1$ and $\sum a_i/(n+1)=a$. We want to prove that $\sum p_i=0$.
We know that for $i \neq j$: \begin{align} \|p_i-a\|^2 &= \|p_i-a_j\|^2\\ (p_i,a_j) &= (p_i,a)+\frac{\|a_j\|^2-\|a\|^2}{2}. \end{align} Summing up over all $j\ne i$: \begin{align} (p_i,(n+1)a-a_i) &= n(p_i,a)+\frac{n-n\|a\|^2}{2} \\ (p_i,a_i) &= (p_i,a)+\frac{-n+n\|a\|^2}{2} \\ \sum_i(p_i,a_j) &= (p_j,a_j)+\sum_{k\ne i}(p_k,a_j)=\sum_i (p_i,a). \end{align} Now $\sum_i p_i$ is orthogonal to $a-a_i$, but these vectors generate the whole space and we conclude that $\sum_i p_i=0$.
This is a special case of the "circumcenter of mass" theorem. See the Wikipedia article and
Akopyan, Arseniy V., Some remarks on the circumcenter of mass, Discrete Comput. Geom. 51, No. 4, 837-841 (2014). ZBL1301.51022.
