E, f, g and h are the midpoint of CD, AC, ao and od respectively.
Making auxiliary lines AD, OC
From the midline theorem, we know that GF = He and Hg = EF.
It is easy to prove that a congruent triangles ABD=CBO.
So AD=CO
That is, the midpoint line is a diamond.
The angle BCO=DAB=HGO is also obtained from the congruence formula.
Because the angle BCO+BOC = 90° and BOC = BGF.
So the angle HGF=HGB+BGF=90.
So it is a square.