Reroute

Label the figure as shown. Let the orange angle be $\theta$ and the length of the blue line $BC = 1$ . Then $BE = \sin \theta$ . Since $BG=GE$ , $GE = \dfrac 12 \sin \theta$ , $FE = \dfrac {GE}{\tan \theta} = \dfrac {\sin \theta}{2\tan \theta} = \dfrac 12 \cos \theta$ . $FC = FE+EC$ $= \dfrac 12 \cos \theta + \cos \theta$ $= \dfrac 32 \cos \theta$ . Let $H$ be the midpoint of $FC$ . Then $FH = \dfrac 34 \cos \theta$ . Since $\triangle CDF$ is isosceles, $DH$ is perpendicular to $FC$ . The length of yellow line $DF = \dfrac {FH}{\cos \theta} = \dfrac 34$ . Then $\dfrac {DF}{BC} = \boxed{\dfrac 34}$ , since $BC=1$ .