How can I prove that DEF is an equilateral triangle?

Question

TRIANGLE ABC IS EQUILATERAL AND

I drawed the picture fast but, AD=BE=CF

(Large Version)

With this information I should be able to prove that DEF is an equilateral triangle. Could anyone help me out?

Answer 1

As the original triangle is equilateral we see that the segments $\overline {DB},\overline {EC},\overline {AF}$ are all congruent. Thus the three triangles $\Delta DBA,\Delta ECF,\Delta FAD$ are congruent (By Side-Angle-Side) and we are done.

Answer 2

This is not enough. What about the other segments of the outer triangle sides, like $DB$?

Your drawing suggests that $ABC$ is a triangle as are $DBE$ and so on, but you have to state this as condition, if you do not want your reader to cherry pick what he likes best from the sketch.

Answer 3

I am sure this is incomplete

More information is needed

Only if ABC IS EQUILATERAL IT IS POSSIBLE.

If it is equlateral

we see rhat

The three triangles $\Delta DBA,\Delta ECF,\Delta FAD$ are congruent (By Side-Angle-Side) and we are done.