Let H be the orthocentre of \(\triangle ABC\) for \(A (0,0) \),\(B(8,0)\),\(C (4,4\sqrt { 3 } )\)

If \(G_1, G_2, G_3, \ldots \) are centroids of triangles \( \triangle AHB, \triangle AG_1 B, \triangle A G_2 B \ldots \) respectively.

Find the value of \(\displaystyle{\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \text{Area}(\Delta A{ G }_{ k }B) } } }\).

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