TITLE: Is ABC an equilateral triangle
QUESTION [22 upvotes]: In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?

REPLY [14 votes]: Not if DAB etc are not colinear. Consider the following:

However, if they are, the centre two triangles in this give a visual proof:

(added as visual complement to Quang Hoang's answer above)