Triangle Hexagon

Author:
kdcampe
Triangle ABC is equilateral and segments DH, EG, and IF are each parallel to a side of the triangle. Move points D, E, and F and observe. What do you notice about polygon DGEIFH? Is it always a hexagon? --If not, when is it a hexagon? --If not, what other shapes can be created? Why are its angles the size(s) they are? Thanks to Ed Southall @solvemymaths for inspiration.