Problem 1431

1. Identify the given elements and properties:
- We have a parallelogram $ABCD$.
- Equilateral triangles $ABE$ and $BCF$ are constructed externally on sides $AB$ and $BC$ respectively.
- We need to prove that $\triangle DEF$ is equilateral.

2. Use properties of parallelograms and equilateral triangles:
- In a parallelogram, opposite sides are equal and parallel. Therefore, $AD = BC$ and $AB = CD$.
- Since $\triangle ABE$ is equilateral, $AE = EB = AB$.
- Similarly, since $\triangle BCF$ is equilateral, $BF = BC = CF$.

3. Analyze angles:
- Let $\angle BAD = \theta$. Since $ABCD$ is a parallelogram, $\angle DAB = \theta$ and $\angle ABC = 180^\circ - \theta$.
- In $\triangle ABE$, $\angle AEB = 60^\circ$ because it is an equilateral triangle.
- In $\triangle BCF$, $\angle BFC = 60^\circ$ because it is an equilateral triangle.

4. **Calculate angles involving points $D$, $E$, and $F$:**
- $\angle DAE = \angle DAB + \angle BAE = \theta + 60^\circ$.
- $\angle EBF = 360^\circ - (\angle ABC + \angle BCF + \angle CBF) = 360^\circ - (180^\circ - \theta + 60^\circ + 60^\circ) = 60^\circ + \theta$.

5. Prove congruence of triangles:
- $\triangle EAD$ and $\triangle EBF$ share the angle $\angle EAD = \angle EBF = 60^\circ + \theta$.
- Since $AD = BF$ and $AE = EB$, by the Side-Angle-Side (SAS) criterion, $\triangle EAD \cong \triangle EBF$.

6. Conclude equal sides:
- From the congruence, $ED = EF$.

7. **Analyze angles in $\triangle DEF$:**
- Let $\angle DEA = \alpha$ and $\angle FEB = \beta$.
- Since $\triangle EAD \cong \triangle EBF$, $\alpha = \beta$.
- Therefore, $\angle DEF = 60^\circ - \alpha + \alpha = 60^\circ$.

8. **Conclude that $\triangle DEF$ is equilateral:**
- Since all sides $ED = EF$ and $\angle DEF = 60^\circ$, $\triangle DEF$ is equilateral.

$\blacksquare$