10. For $\triangle A B C$, the lengths of its sides are denoted as $\alpha, \beta, \gamma$, and the distances from its centroid to each vertex are denoted as $\lambda, \mu, v$. Find the value of $\frac{\alpha^{2}+\beta^{2}+\gamma^{2}}{\lambda^{2}+\mu^{2}+v^{2}}$.
10. Original expression $=\frac{\sum\left|z_{A}-z_{_{B}}\right|^{2}}{\sum\left|\frac{1}{3}\left(z_{A}+z_{B}+z_{C}\right)-z_{1}\right|^{2}}$ Let's assume $z_{A}=x_{1}+y_{1} \mathrm{i} \quad z_{B}=x_{2}+y_{2} \mathrm{i} \quad z_{C}=x_{3}+y_{3} \mathrm{i}$, so the original expression $=\frac{2\left(\sum x_{i}^{2}+\sum y_{i}^{2}-2 \sum x_{i} x_{j}-2 \sum y_{i} y_{j}\right)}{\frac{1}{9}\left(6 \sum x_{i}^{2}+6 \sum y_{i}^{2}-6 \sum x_{j} x_{i}-6 \sum y_{i} y_{j}\right)}=\frac{2 \times 2}{\frac{1}{9} \times 6}=6$