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DEF is equilateral => ABC is also equilateral?

Source: Tournament of Towns 2007 - Fall - Junior O-Level - P3

September 3, 2011
geometrycircumcirclegeometry proposed

Problem Statement

DD is the midpoint of the side BCBC of triangle ABCABC. EE and FF are points on CACA and ABAB respectively, such that BEBE is perpendicular to CACA and CFCF is perpendicular to ABAB. If DEFDEF is an equilateral triangle, does it follow that ABCABC is also equilateral?
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