KöMaL 2020 B5117

The points $A$, $B$, $C$, $D$ (in this order) lie on the same line. On the same side of the line, a regular triangle is drawn on each of the line segments $AB$, $BC$ and $CD$, with the third vertices being $E$, $F$, and $G$, respectively. Let the distances between the adjacent points on the line be $AB=a$, $BC=b$, $CD=c$. Show that the measure of $\angle EFG$ equals $120^{\circ}$ if and only if $a+c=b$ or $\frac1a+\frac1c=\frac1b$.