Consider a triangle ABC whose vertices are A (0, α, α), B (α, 0, α) and C (α, α, 0), α>0. Let D be a point moving on the line x+z-3=0=y and G be the…

Consider a triangle ABC whose vertices are $\displaystyle \mathrm{A}(0, \alpha, \alpha), \mathrm{B}(\alpha, 0, \alpha)$ and $\displaystyle \mathrm{C}(\alpha, \alpha, 0), \alpha>0$. Let D be a point moving on the line $\displaystyle x+z-3=0=y$ and G be the centroid of $\displaystyle \triangle \mathrm{ABC}$. If the minimum length of GD is $\displaystyle \sqrt{\frac{57}{2}}$, then $\displaystyle \alpha$ is equal to \_\_\_\_.