6 In the isosceles trapezoid $A B C D$, $A B / / C D$, and $A B > C D$. Let the eccentricity of the hyperbola with foci at $A$ and $B$ and passing through point $D$ be $e_{1}$, and the eccentricity of the ellipse with foci at $C$ and $D$ and passing through point $A$ be $e_{2}$, then $e_{1} e_{2}=$ $\qquad$.
6. As shown in the figure, let $A B=x, C D=y, A D=B C=z, A C=B D=t$, then
$$
\begin{array}{l}
e_{1}=\frac{A B}{B D-A D}=\frac{x}{t-z}, \\
e_{2}=\frac{C D}{A D+A C}=\frac{y}{t+z},
\end{array}
$$

Therefore, $e_{1} e_{2}=\frac{x y}{t^{2}-z^{2}}$. Draw $D H \perp A B$ at $H$,
by the symmetry of the isosceles trapezoid, we get
$$
\begin{aligned}
t^{2}-z^{2} & =B D^{2}-A D^{2} \\
& =B H^{2}-A H^{2} \\
& =(B H+A H)(B H-A H) \\
& =A B \cdot C D \\
& =x y .
\end{aligned}
$$

Therefore, $e_{1} e_{2}=\frac{x y}{t^{2}-z^{2}}=\frac{x y}{x y}=1$.