No. If [math]\overline{AB}\cong\overline{AC}[/math], then the angle bisector will be parallel to [math]\overleftrightarrow{BC}[/math]. In this case, since [math]\frac{AB}{AC}=\frac{1}{1}[/math], it would be impossible for [math]\frac{DB}{DC}=\frac{AB}{AC}[/math], since either [math]DB>DC[/math] or [math]DC>DB[/math], depending on which side the angle bisector intersects [math]\overleftrightarrow{BC}[/math]. In either case, they can never be equal.