6. As shown in Figure 2, in trapezoid $A B C D$, $D C / / A B$, a line $M N$ $/ / A B$ is drawn through the intersection of $A C$ and $B D$, with points $M$ and $N$ on $A D$ and $B C$ respectively. The relationship satisfied by $A B$, $D C$, and $M N$ is ( ).
(A) $A B+D C=\sqrt{5} M N$
(B) $A B \cdot D C=M N^{2}$
(C) $\frac{A B}{M N}-1=\frac{M N}{D C}-1$
(D) $\frac{1}{A B}+\frac{1}{D C}=\frac{2}{M N}$
6.D.

Let the intersection point of $A C$ and $B D$ be $O$.
By the properties of similar triangles, we have
$$
\frac{M O}{A B}=\frac{D O}{D B}=\frac{C O}{C A}=\frac{O N}{A B} \text {. }
$$

Therefore, $M O=O N, M N=2 M O$.
Also, $\frac{M O}{D C}+\frac{M O}{A B}=\frac{A M}{A D}+\frac{D M}{A D}=1$, thus
$$
\frac{1}{A B}+\frac{1}{D C}=\frac{1}{M O}=\frac{2}{2 M O}=\frac{2}{M N} \text {. }
$$