It is obvious if you think about it as follows:
The opposite sides are parallel. If you first move the two parts of the hexagon parallel to one pair of opposite sides until either $a=0$ or $d=0$, then move parallel to another pair until either $b=0$ or $e=0$, and then do the same with the third pair until either $c=0$ or $f=0$, then the remainder is just an equiangular, hence equilateral triangle and we are done.
Of course one can do this more formally with vectors or complex numbers but this somewhat obscures the geometric nature of the problem.
