Question

In the adjoining figure, ABCD is a parallelogram and X, Y are the points on diagonal BD, such that DX = BY. Prove that CXAY is a parallelogram.

Answer 1

Given : ABCD is a parallelogram.
To prove: CXAY is a parallelogram.
Construction : Join AC, meeting BD at O.
Proof: We know that the diagonals of a parallelogram bisect each other.
So, OA = OC and OD = OB.
OD = OB...(i)
DX = BY.........(ii)
Subtracting (ii) from (i),we get:
$\Rightarrow$OD$-$DX = OB$-$BY
$\Rightarrow$OX = OY

Since, CXAY is a quadrilateral whose diagonals bisect each other, it is a parallelogram.