vectors(2)

BC=2BF
BD=2BE

CD=BD-BC
=2BE-2BF
=2(BE-BF)
=2FE

By similar argument:
AD=2AH
AC=2AG

CD=AD-AC
=2AH-2AG
=2(AH-AG)
=2GH

2FE=2GH
FE=GH

FE=GH and FE//GH
Therefore EFGH is a parallelogram.