Inscribed Equilateral Hexagons

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In any triangle, there exists a continuous family of convex equilateral hexagons inscribed in such a way that each side of the triangle is flush with at least one of the hexagon's sides. In some cases, the equilateral hexagon degenerates to a pentagon, quadrilateral, or triangle.

For more information, see Inscribing Equilateral Polygons in Triangles, by Milena Harned and Miriam Rittenberg, in the Girls' Angle Bulletin, Volume 11, Number 2.