BALKAN-SL 2014 G4

Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$ . We consider the excenters $A_1$ , $B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$ ,of the triangle $A_1B_1C_1$ . By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.