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.