8,9
[ Ratio in which the bisector divides the side ]

In an isosceles triangle $A B C$, a rhombus $D E C F$ is inscribed such that vertex $E$ lies on side $B C$, vertex $F$ on side $A C$, and vertex $D$ on side $A B$. Find the length of the side of the rhombus if $A B=B C=12, A C=6$.
Let $x$ be the side of the rhombus. Since $D E \| A C$, triangles $D B E$ and $A B C$ are similar. Therefore, $B E: B C=D E: A C$, or $12-x / 12=x / 6$. From this, $x=4$.

## Answer

4.