4. In an equilateral triangle $A B C$ the side is equal to $a$. On side $B C$ lies point $D$, and on $A B$ point $E$ such that $|B D|=\frac{1}{3} a,|A E|=|D E|$. Find the length of $C E$.
4. If $|E D|=|A E|=x$, then $|B E|=a-x$. Writing the cosine theorem for $\triangle B D E$:

$$
x^{2}=\frac{a^{2}}{9}+(a-x)^{2}-\frac{2}{3} a(a-x) \cdot \frac{1}{2}
$$

we find $x=\frac{7}{15} a$.

Answer: $: C E \left\lvert\,=\frac{13}{15} a\right.$.