Question 1065070
Well-slept, on a Saturday morning, I have an interpretation and answer to that problem.
If you enter a thank-you note, could you tell me where the problem came from, and if my reading/interpretation of the problem is correct?
The segment lengths are {{{highlight(system(BX=CY=3in,XY=2in))}}} .


The text posted does not give enough information to identify what  parallelogram ABCD looks like.
Is there a typo? Is it a poorly designed problem?
Assuming that it was a problem designed by an intelligent mind,
it took a less sleep-deprived mind to decipher what interpretation/reasoning/answer might be expected.
 
Below are two parallelograms that could be ABCD.
I did not label vertices, because it does not matter:
AD is one of the long sides, and AB is one of the short sides.
{{{drawing(1100,600,-0.5,10.5,-2.4,3.6,
line(0,0,8,0),line(2.12,-2.12,10.12,-2.12),
line(2.12,-2.12,0,0),line(8,0,10.12,-2.12),
line(5.12,-2.12,3,0),line(7.12,-2.12,5,0),
line(0,0.5,8,0.5),line(1.03,3.32,9.03,3.32),
line(1.03,3.32,0,0.5),line(8,0.5,9.03,3.32),
line(4.03,3.32,3,0.5),line(6.03,3.32,5,0.5),
arrow(5,0.25,8,0.25),arrow(3,0.25,0,0.25),
locate(3.7,0.35,8),locate(4,0.35,inches),
locate(8.6,1.9,3in),locate(0.85,-1.1,3in),
locate(1.4,0,3in),locate(3.9,0, 2in),locate(6.4,0,3in)
)}}}
I split the parallelograms into 3 parallelograms.
The ones at the ends (where A and D are) are rhombi,
and the angle bisectors, contain AX and DY,
are diagonals of those rhombi.