12. Let three distinct complex numbers $z_{1}, z_{2}, z_{3}$ satisfy the equation $4 z_{1}^{2}+5 z_{2}^{2}+5 z_{3}^{2}=4 z_{1} z_{2}+6 z_{2} z_{3}+4 z_{3} z_{1}$. Denote the lengths of the sides of the triangle formed by $z_{1}, z_{2}, z_{3}$ on the complex plane, in ascending order, as $a, b, c$, then $a: b: c=$ $\qquad$.
$$
\begin{array}{l}
4 z_{1}^{2}+5 z_{2}^{2}+5 z_{3}^{2}=4 z_{1} z_{2}+6 z_{2} z_{3}+4 z_{3} z_{1} \\
\Rightarrow 4\left(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{1} z_{2}-z_{2} z_{3}-z_{3} z_{1}\right)+\left(z_{2}-z_{3}\right)^{2}=0 \\
\Rightarrow 2\left(z_{1}-z_{2}\right)^{2}+3\left(z_{2}-z_{3}\right)^{2}+2\left(z_{3}-z_{1}\right)^{2}=0
\end{array}
$$

Let $u=z_{1}-z_{2}, v=z_{2}-z_{3}, w=z_{3}-z_{1}$,
$$
\begin{array}{l}
\text { then }\left\{\begin{array}{l}
u+v+w=0, \\
2 u^{2}+3 v^{2}+2 w^{2}=0
\end{array} \Rightarrow 2 u^{2}+3 v^{2}+2(u+v)^{2}=4 u^{2}+5 v^{2}+4 u v=0\right. \\
\Rightarrow 4\left(\frac{u}{v}\right)^{2}+4 \cdot \frac{u}{v}+5=0 \Rightarrow \frac{|u|}{|v|}=\left|\frac{u}{v}\right|=\sqrt{\frac{u}{v} \cdot \overline{\left(\frac{u}{v}\right)}}=\frac{\sqrt{5}}{2} \text {, similarly we get } \\
2 u^{2}+3(u+w)^{2}+2 w^{2}=5 u^{2}+5 w^{2}+6 u w=0 \Rightarrow 5\left(\frac{u}{w}\right)^{2}+6 \cdot \frac{u}{w}+5=0 \Rightarrow \frac{|u|}{|w|}=1 \text {. } \\
\end{array}
$$

Therefore, $|u|:|v|:|w|=\sqrt{5}: 2: \sqrt{5} \Rightarrow a: b: c=2: \sqrt{5}: \sqrt{5}$.