Brazilian Mathematical Olympiad

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 $\vec{GE}$ is represented by the complex number $h = \frac{a+b-c-d}{2} + i\frac{-a+b+c-d}{2}$ and the vector $\vec{FH}$ 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.