6. Given that the incircle $\odot O$ of quadrilateral $ABCD$ touches $AB$, $BC$, $CD$, and $DA$ at points $E$, $F$, $G$, and $H$ respectively, and $AF$, $DF$ intersect $EG$ at points $M$ and $N$. If $BF=CF=5$, $EG=6$, then the length of $MN$ is ( ).
(A) 3
(B) 4
(C) 5
(D) an indeterminate value
6. A.

As shown in Figure 3, connect $EF$, $GF$, and $HF$. Draw a line through point $A$ parallel to $EG$, intersecting lines $EF$ and $FH$ at points $P$ and $Q$, respectively. $HF$ intersects $EG$ at point $T$.