Ponder This Challenge - August 1998 - Equilateral triangle

We have a triangle ABC, with a point D on side AB, E, on side BC, and F on side CA. The smaller triangle, DEF, is equilateral. The line segments AD, BE, and CF all have equal length.
Problem: Prove that ABC is also equilateral.

Comment: Although we have a solution for this problem, it is not a simple one.

Solution

• Click here to view the solution

  2005: Fuxiang Yu sent in a much improved solution to this puzzle:

  Read Fuxiang Yu's solution (89KB)

  2012: Zhong Nan and Sandy Jelly sent another simpler solutions:

  Read Zhong Nan's solution (114KB)

  Read Sandy Jelly's solution (150KB)

  2016: Alek Vainshtein sent another nice solution:

  Read Alek Vainshtein's solution (24KB)

  2020: Yang Haibo, a physics teacher from China, sent us yet another solution:
  

  Most of the submited solutions demonstrated the fundamental challenge of this problem: How do you prove something that seems obviously true? The problem’s simplicity and the intuitive “obviousness” of the conclusion lead many responders down dead end paths. Some suggested using trigonometric functions to prove triangle ABC equilateral, but none of the other triangles present are implicitly “right” triangles. Others began with the assumption that triangles ADF, BED, and CFE were congruent, which cannot be assumed, but rather needs to be proved.
  Here is our solution. Click for the IBM techexplorer version.

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