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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?

Perhatikan gambar berikut. Titik D dan E berada pada garis AB, dan Titik F berada pada garis AC sehingga DA = DF = DE, BE = EF dan BF = BC. Diketahui ∠ABC = 2∠ACB. Jika ∠BFD = x°. Tentukan nilai x.

Solve a geometry problem involving points on a triangle, where ∠ABC = 2∠ACB and other relations between segments are given. The solution involves properties of isosceles triangles and angle bisectors. Suitable for students in Grades 9-11.

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