Prove:

Judging from the problem:

Angle EAD= Angle DAPFAB= BAP, FAE= EAD+ DAP+ FAB+ BAP= 180 degrees.

2 (angle DAP+ angle PAB)= 180 degrees; Angle DAB=90 degrees.

It can also be proved that angle ADC= angle DCB= angle CBA= angle BAD=90 degrees.

So the quadrilateral ABCD is a rectangle.

AD= root 3; AB= root 6; Therefore, BD=3.

Angle F= angle AFB= angle H= angle CQD, so AP is parallel to CQ.

Angle ABD= angle CDB；; Angle APB= angle CQD；; AB=DC

Triangle CDQ

BP = QD = BF = DH； DP=BQ=DE=BC

BD=PD+PB=PD+QD=ED+DH=EH=3

Area: 2(ABxAD)=3

Side length: 4EH= 12

end