Three. (20 points) As shown in Figure 2, the distance from the center O to a chord of circle $\odot O$ is $O H=2008$. In the two segments, two squares $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ are constructed, where points $A, D, A_{1}, B_{1}$ are on the arc, and points $B, C, C_{1}, D_{1}$ are on the chord. Find the difference in the side lengths of these two squares (the larger side minus the smaller side).
Three, as shown in Figure 5, let the side lengths of the squares $A_{1} B_{1} C_{1} D_{1}$ and $A B C D$ be $a$ and $b$, respectively. Extend $O H$ to intersect $A_{1} B_{1}$ at point $M$, and draw $O N \perp C D$ at point $N$.
By the Pythagorean theorem, we have
$$
\begin{array}{l}
O M^{2}+M A_{1}^{2}=R^{2} \\
=O N^{2}+D N^{2} . \\
\text { Therefore, } (2008+a)^{2}+\left(\frac{a}{2}\right)^{2} \\
=\left(\frac{b}{2}\right)^{2}+(b-2008)^{2}, \\
2 \times 2008(a+b)=\frac{5}{4}(a+b)(b-a) .
\end{array}
$$

Thus, $b-a=\frac{8}{5} \times 2008=3212.8$.