JEE Math Practice Question

Three lines are given by \[ \vec{r} & =\lambda \hat{i}, \lambda \in \mathbb{R} \] \[\vec{r} & =\mu(\hat{i}+\hat{j}), \mu \in \mathbb{R} \] \[ \vec{r} =v(\hat{i}+\hat{j}+\hat{k}), v \in \mathbb{R}. \] Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then what is the value of $(6 \Delta)^{2}$?