19. $A B C$ and $G B D$ are straight lines. $E$ is a point on $C D$ produced, and $A D$ meets $E G$ at $F$. If $\angle C A D=\angle E G D, E F=F G, A B: B C=1: 2$ and $C D: D E=3: 2$, find $B D: D F$.
(3 marks)
$A B C$ 和 $G B D$ 均為直線, $E$ 是 $C D$ 延線上的一點, 且 $A D$ 交 $E G$ 於 $F$ 。若 $\angle C A D=$ $\angle E G D, E F=F G, A B: B C=1: 2$, 且 $C D: D E=3: 2$, 求 $B D: D F$ 。
19. $\sqrt{3}: 2$
19. Let $[D E F]=2$.
Since $C D: D E=3: 2,[F C D]=3$.
Since $A B: B C=1: 2,[F B D]=1$.
Since $E F=F G,[G F D]=[D E F]=2$.
So $[G B F]=1=[F B D]$, i.e. $G B=B D$
Together with $G F=F E, B F / / C E$.
Hence $A F: F D=A B: B C=1: 2$.
Let $G B=B D=x, A F=y, F D=2 y$.

Since $\angle C A D=\angle E G D, G A F B$ is a cyclic quadrilateral.
Thus $D B \times D G=D F \times D A$, i.e. $x(2 x)=2 y(3 y)$.
As a result we have $x: y=\sqrt{3}$.
It follows that $B D: D F=x: 2 y=\sqrt{3}: 2$.