Show that $(F,G_1;H,B)$ is an harmonic range in the figure below
The complete quadrangle $DEXY$ has the property that one pair of opposite sides intersect at $B$, a second pair intersect at $G_1$, and the third pair meet $BG_1$ at $F, H$. Therefore, $FG_1HB$ is a harmonic range—indeed, the existence of such a quadrangle is often taken as the definition of a harmonic range.
This immediately proves the statement in your question title, but of course it ignores most of your diagram—which doesn’t make it wrong, but does make me wonder whether you’re working with a different definition.