A3. Triangle $A B C$ is divided into 4 smaller triangles $A D E, D F E$, $E F C$ and $F B C$ with equal areas (see figure). What is the ratio of the lengths $|A F|:|D B|$?
(A) $1: 1$
(B) $9: 8$
(C) $8: 7$
(D) $7: 6$
(E) $6: 5$

![](https://cdn.mathpix.com/cropped/2024_06_07_716acb2f660115fb0b44g-10.jpg?height=280&width=459&top_left_y=1536&top_left_x=1461)
A3. Triangles $A E D$ and $D E F$ have areas in the ratio 1:1 and share a common height through vertex $E$, so their sides are also in the ratio $|D A|:|F D|=1: 1$. Triangles $A C B$ and $F C B$ have areas in the ratio 4:1 and share a common height through vertex $C$, so their sides are in the ratio $|B A|:|B F|=4: 1$. Therefore, $|B A|=4|B F|$, $|F A|=|B A|-|B F|=3|B F|$, $|F D|=\frac{1}{2}|F A|=\frac{3}{2}|B F|$, and $|B D|=|B F|+|F D|=\frac{5}{2}|B F|$. It follows that $|F A|:|B D|=3|B F|: \frac{5}{2}|B F|=6: 5$.