**Given :** ABCD is a quadrilateral in which EG and FH are the line segments joining the mid-points of opposite sides.

**To Prove :** EG and FH bisect each other.

**Construction :** Join EF, FG, GH, HE and AC.

**Proof :** In ∆ABC, E and F are mid-points of AB and BC respectively.

∴ EF = 1/ 2 AC and EF || AC …(i)

In ∆ADC, H and G are mid-points of AD and CD respectively.

∴ HG = 1 2 AC and HG || AC …(ii)

From (i) and (ii), we get

EF = HG and EF || HG

∴ EFGH is a parallelogram. [∵ a quadrilateral is a parallelogram if its one pair of opposite sides is equal and parallel]

Now, EG and FH are diagonals of the parallelogram EFGH. ∴ EG and FH bisect each other.

[Diagonal of a parallelogram bisect each other] Proved.