CEMC Hypatia, 2011

In the diagram, $D$ and $E$ are the midpoints of $AB$ and $BC$ respectively. (The diagram shows a coordinate plane with $B(0,0)$, $A(0,6)$, and $C(8,0)$; $D$ is the midpoint of $AB$, $E$ is the midpoint of $BC$, segments $AE$ and $CD$ are drawn, and $F$ is their point of intersection.)
(a) Determine an equation of the line passing through the points $C$ and $D$.
(b) Determine the coordinates of $F$, the point of intersection of $AE$ and $CD$.
(c) Determine the area of $\triangle DBC$.
(d) Determine the area of quadrilateral $DBEF$.