 Solution

All roots of $A_3$ have the same length, so $|{\alpha }_2|/|{\alpha }_3|=1$. In $B_3$, ${\alpha }_2$ is long and ${\alpha }_3$ short, giving $\sqrt{2}$. In $C_3$ their roles are reversed, giving $1/\sqrt{2}$. These ratios also follow from $A_{23}/A_{32}=|{\alpha }_2{|}^2/|{\alpha }_3{|}^2$.