Inspired by Centroid Circle: a problem by Emmanuel Lasker

Let $ABC$ be a triangle, with $AB=11$ and let $D,E,F,G$ respectively be the midpoint of the side $AC$ , the midpoint of $AB$ ,the midpoint of $BC$ and the centroid of $ABC$ . Knowing that $A,D,E,G$ as well as $B,E,G,F$ are concyclic, and that $BC=\sqrt{n}$ , Find $\Sigma n$ i.e. the sum of all possible values of $n$ .