Dutch Mathematical Olympiad

We have a triangle $\triangle ABC$ and two points $D$ and $E$ on the line segment $BC$, such that $\angle BAD = \angle DAE = \angle EAC$. Furthermore, $|AB| = 6$, $|BD| = 4$, $|DE| = 2$, and $|EA| = 3$. See the picture on the right: be aware, it is not drawn to scale!

What is the length of the line segment $CE$?
A) 2 B) $2\frac{1}{4}$ C) $2\frac{1}{2}$ D) $2\frac{3}{4}$ E) 3