Parallelograms in a convex hexagon

Question

If you have a (convex) hexagon, and label it ABCDEF, and if ACDF is a parallelogram, and ABDE is a parallelogram, prove that BCEF is a parallelogram also.

Answer 1

Choose an origin $O$. Opposite sides of a parallelogram are equal vectors. You therefore have the vector equations:

$OC-OA=OD-OF$ and $OD-OC=OF-OA$ from the first, equivalent to $OC+OF=OA+OD$

$OB-OA=OD-OE$ and $OD-OB=OE-OA$ from the second, equivalent to $OB+OE=OA+OD$

And for the third you need $OC-OB=OE-OF$ and $OE-OC=OF-OB$, equivalent to $OC+OF=OB+OE$