DE-BUNDESWETTBEWERB 2001 Problem 7

Let $ABC$ be a triangle. Points $A',B',C'$ are on the sides $BC, CA, AB,$ respectively such that we have $$\overline{A'B'} = \overline{B'C'} = \overline{C'A'}$$ and $$\overline{AB'} = \overline{BC'} = \overline{CA'}.$$ Prove that triangle $ABC$ is equilateral.