The problem describes a geometric figure where we need to determine the value of $x$. Here's the given information and how we can approach the solution:

Given:

1. $D$ and $E$ lie on the line $AB$, and $F$ lies on the line $AC$.
2. $DA = DF = DE$, meaning $D$ is equidistant from points $A$, $F$, and $E$.
3. $BE = EF$, and $BF = BC$.
4. $\angle ABC = 2 \times \angle ACB$.
5. $\angle BFD = x^\circ$.

Approach:

1. From $\angle ABC = 2 \times \angle ACB$, we infer that $\triangle ABC$ is an isosceles triangle where $AB = AC$. This symmetry might simplify angle relations.
2. Using the fact that $D$ is equidistant from $A$, $F$, and $E$, and other distance conditions (like $BE = EF$ and $BF = BC$), we may have certain symmetry or specific triangle properties to apply.
3. The goal is to find $x$, the measure of $\angle BFD$.

Let me process the geometric configuration to determine the value of $x$.From the given information and the setup of the problem, we deduce that:

$x = 2 \times y$

Where $y$ represents $\angle ACB$. Since $\angle ABC = 2 \times \angle ACB$, the angle $x = 2 \times \angle ACB$, meaning $x$ depends directly on the value of $\angle ACB$.

This suggests that $x$ is twice the value of the angle at $\angle ACB$. If any additional values for the angles in the triangle are provided, we can substitute them to get a numerical value for $x$.

Would you like to explore more detailed steps, or do you have any specific angle values you would like to use?