ATRACTOR

Build the point \(H\) over the line segment \([AB]\) such that \(\overline{BH} = \overline{BD}\) and the point \(I\) over the line segment \([AC]\) such that \(\overline{CI} = \overline{CD}\). As \(D \hat{B}F = H \hat{B}F\), the triangles \([DBF]\) and \([HBF]\) are congruent, whereby \(\overline{HF} = \overline{FD}\). Analogously, the triangles \([DCG]\) and \([ICG]\) are also congruent, and \(\overline{DG} = \overline{GI}\). Furthermore, as \([FDG]\) is an equilateral triangle, \(\overline{FD} = \overline{FG} = \overline{DG}\). Hence, \(\overline{HF} = \overline{FG} = \overline{GI}\).

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