TITLE: Parallelograms in a convex hexagon
QUESTION [0 upvotes]: 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.

REPLY [0 votes]: 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 $OC+OF=OB+OE$
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