This can be proved graphically.
A(0,0), B(b₁,b₂), C(c₁,c₂), D(d₁,0) form a quadrilateral ABCD.
The midpoint of a line has the average of the coordinates of its endpoints.
The midpoint of AB is P(b₁/2,b₂/2),
midpoint of BC is Q((b₁+c₁)/2,(b₂+c₂)/2),
midpoint of CD is R((c₁+d₁)/2,c₂/2),
midpoint of AD is S(d₁/2,0),
midpoint of PR is X((b₁+c₁+d₁)/4,(b₂+c₂)/4),
midpoint of QS is X((b₁+c₁+d₁)/4,(b₂+c₂)/4).
So the midpoint of PR and QS is the same (point X), therefore the line joining the midpoints of opposite sides of a quadrilateral bisect one another.