Inspired by Centroid Circle: a problem by Emmanuel Lasker

Geometry Level 4

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 \(n\).

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