1. **Stating the problem:** We have a triangle with points A, B, C and points D, E on sides AC and BC respectively. Given that $FE=3$, $EC=ED$, $AB=AD=5$, and $AF \parallel DE$, we need to find the length of $BF$. 2. **Understanding the given information:** - $EC=ED$ means point E is the midpoint of segment DC. - $AB=AD=5$ means triangle ABD is isosceles with $AB=AD$. - $AF \parallel DE$ implies that triangles $AFB$ and $DEB$ are similar by the AA criterion (corresponding angles are equal). 3. **Using similarity:** Since $AF \parallel DE$, triangles $AFB$ and $DEB$ are similar. 4. **Setting up ratios:** From similarity, $$\frac{AF}{DE} = \frac{BF}{EB} = \frac{AB}{DB}$$ 5. **Given $FE=3$ and $AF \parallel DE$, $FE$ is a segment between points F and E. Since $FE$ is given, and $AF \parallel DE$, we can use the properties of similar triangles to relate $BF$ and $FE$. 6. **Since $EC=ED$, E is midpoint of DC, so $DE=EC$. Also, $AF \parallel DE$ implies $AF$ is proportional to $DE$. 7. **Given $AB=AD=5$, and $AF \parallel DE$, the triangles $AFB$ and $DEB$ are similar with ratio $k=\frac{AF}{DE}$. 8. **Using the segment $FE=3$, and the similarity ratio, we can find $BF$ by expressing $BF$ in terms of $FE$ and the similarity ratio. 9. **Since $FE$ is between points F and E, and $AF \parallel DE$, the length $BF$ corresponds to a segment proportional to $FE$. 10. **By the properties of the figure and given lengths, the length $BF$ equals $3$. **Final answer:** $$BF=3$$