VI OBM

Consider $A,B,C,D$ to be the points $a,b,c,d$ in the complex plane. Then $E,F,G,H$ are the points $\frac{a+b}{2}+i\frac{b-a}{2}$, $\frac{b+c}{2}+i\frac{c-b}{2}$, $\frac{c+d}{2}+i\frac{d-c}{2}$, $\frac{d+a}{2}+i\frac{a-d}{2}$. Hence vector $\overrightarrow{GE}$ is represented by the complex number $h=\frac{a+b-c-d}{2}+i\frac{-a+b+c-d}{2}$ and the vector $\overrightarrow{FH}$ by the $k=\frac{a-b-c+d}{2}+i\frac{a+b-c-d}{2}$. But $k=ih$, so $EG$ and $FH$ are equal and perpendicular.