How to prove that this defined language is regular? Suppose that $L$ is a regular language and $0&lt;\alpha&lt;1$ and $\alpha \in Q$. Define $L_\alpha$ as

$$L_\alpha = \{\omega \in \Sigma^* \mid \exists \omega_1 \in  \Sigma^* .\omega\omega_1 \in L,\frac{|\omega|}{|\omega\omega_1|}=\alpha\}$$

How do I prove that $L_\alpha$ is regular?

Here is my attempt. If we assume that $D$ is the automaton that recognize $L$ with $Q=\{q_0,...,q_n\}$, if $\delta(q_0,\omega\omega_1)\in F$ and $\delta(q_0,\omega)=q_i$  we can define  $D_1$ such that $\delta_1(q_0,\omega)=r_{i-1}$ (for $r_i \in F_1)$, but the problem is I can't define $\delta_1(q_0,a) $for $a\in \Sigma$.
Try proving it for $\alpha = 1/2$ first.  Construct a non-deterministic automaton that recognizes $L_\alpha$.  Before you start reading in the string $w$, guess something about where you will be in the future.  What would you like to guess?  What would help you?