We can easily compute the coordinate of each point. Let $B$ be the origin with $C(1.0)$ and $A(a,b)$ So $D(0.-1)$ because it is on the square $BCED$ And $G(1.2b)$ because it is on the square $ACFG$ and because $\angle{ACB}=45°$ so $\angle{BCG}=90$ and so $G$ is on the line $y=1$ So the coordinates of the middle of $DG$ are $H(1/2.b-1/2)$ $H$ is on the line $y=1/2$ which is the mediator of $BC$. Symetricaly we can find that $H$ is on the mediator of $AC$. QED