JEE Type

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