I find proof for this problem on simplex which I think it is true

Question. $A_0A_1...A_n$ is a simplex in $\Bbb E^n.$ $G$ is its centroid and its center circumscribed sphere is $O.$ Let $O_i$ be the centers circumscribed sphere of the simplies $GA_{0}A_{1}...A_{i-1}A_{i+1}...A_n.$ Prove that $O$ is centroid of the simplex $O_0O_1...O_n.$

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 $O_0O_1...O_n.$