12. If an isosceles trapezoid can be divided into two isosceles triangles, it is called a golden trapezoid. The ratio of the four sides of a golden trapezoid is
12. $1: 1: 1: \frac{\sqrt{5}+1}{2}$.

There is only one possibility, as shown in Figure 8.
Draw the angle bisector of $\angle B D C$, intersecting $B C$ at $E$. Then
$\triangle D E C \backsim \triangle B D C$, $B E=E D=C D$.
Let $B E=1, E C=x$, then $\frac{1}{x}=\frac{x+1}{1}$.
Since $x>0$, we have $x=\frac{\sqrt{5}-1}{2}$. Therefore, $B C=\frac{\sqrt{5}+1}{2}$.
Thus, the required ratio is $1: 1: 1: \frac{\sqrt{5}+1}{2}$.