Question

ABCD is a parellelogram, X and Y are points on DC and AB such that AY = CX. Prove that XY and BD bisect each other.

Answer 1

See the diagram.
Let XY and BD intersect at O.

given that AY=CX
we know that AB= CD
so AB- AY = CD- CX
or BY=DX

In triangle, YOB and XOD, 
angle YOB = angle XOD 
angle ODX = angle OBY 
BY=DX

So both are congruent.
Thus OD = BO
and XO = YO
 So they bisect each other.(proved)

Answer 2

we know that opposite sides are equal in parellelogram   
AB=CD
1/2 AB=1/2 CD 
let BD and XY meets at O
in triangle DOX and triangle BOY  DX = by proved 
angle DOX =angle BOY vertically opposite angle 
angle ODX = angle OBY alternate interior angle 
therefore triangle DOX is congurant to triangle BOY therefore DO = BO 
therefore  XY bisects BD 

hence proved