Construct lines KH, BD and GL
We know that K and H are the midpoints of AD and AB
Consider △ ABD
Using the mid-point theorem
We get
KH = ½ BD
Consider △ CBD
Based on the midpoint theorem
GL = ½ BD
So we get
KH = GL
Consider △ KOH and △ GOL
We know that KH = GL
From the figure we know that alternate angles are equal
So we get
∠ OKH = ∠ GLO and ∠ OHK = ∠ OGL
By SAS congruence criterion
△ KOH ≅ △ GOL
OG = OH and OK = OL (c. p. c. t)
So we know that GH and KL bisect each other.
Therefore, it is proved that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.