Example 3. As shown in Figure 3, the equilateral triangles $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ have their sides $A C$ and $A_{1} C_{1}$ bisected at $O$. Then $A A_{1}: B B_{1}=$ $\qquad$
This problem can also be solved using the "specialization" method (let $A_{1}$ be on $AO$). However, if $B O, B_{1} O$ are considered, it is immediately known that $\triangle A A_{1} O$ ○ $\triangle B B_{1} O$, thus $A A_{1}: B O_{1}=A O: B O=\sqrt{3}$ : 3.