Hong Kong Preliminary Selection Contest

Note that $BD$ is the perpendicular bisector of $AC$, while $EG$ is the perpendicular bisector of $CF$. Thus the intersection of $BD$ and $EG$, which we denote by $O$, is the circumcentre of $△ACF$.

As $\angle OBG=\angle OGB=45^{\circ }$, $△OBG$ is isosceles. Since $OH=OC$, we have $BH=CG=567$. It follows that $CH=1234-567=667$.