MathNet 2014 Question 18

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.

Supplementary schematic (AI-generated):