In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that :

(i) AX = YC

(ii) AX is parallel to YC

(iii) AXCY is a parallelogram

(i) We know that,

Opposite sides of || gm are equal.

∴ AB = CD

⇒ $\dfrac{AB}{2} = \dfrac{CD}{2}$

⇒ AX = CY (As, X and Y are mid-points of AB and CD respectively)

Hence, proved that AX = YC.

(ii) We know that,

Opposite sides of || gm are parallel.

∴ AB || DC

∴ AX || YC.

Hence, proved that AX || YC.

(iii) From figure,

AX = YC and AX || YC.

Since, one pair of opposite sides of quadrilateral AXCY are equal and parallel.

∴ AXCY is a || gm.

Hence, proved that AXCY is a parallelogram.