4. Let $E$ be the midpoint of the base $A B$ of trapezoid $A B C D$, where $|A B|:|C D|=$ $=3: 1$. Diagonal $A C$ intersects segments $E D, B D$ at points $F, G$ respectively. Determine the successive ratio

$$
|A F|:|F G|:|G C| \text {. }
$$
SOLUTION. Since the problem statement and question contain only ratios, we can choose the side lengths of the trapezoid as suitable specific numbers. Let's choose, for example, $|AB| = 6$, then $|AE| = |BE| = 3$ and $|CD| = 2$. Let the sought lengths be $|AF| = x, |FG| = y, |GC| = z$. We have marked these lengths on Fig. 1, along with three pairs of congruent angles, which we will now use in our reasoning about similar triangles according to the $uu$ theorem.

Triangles $ABG$ and $CDG$ are similar, so $(x + y) : z = 6 : 2 = 3 : 1$. Triangles $AEF$ and $CDF$ are also similar, so $x : (y + z) = 3 : 2$.

![](https://cdn.mathpix.com/cropped/2024_04_17_842e4c351c34ef14d825g-4.jpg?height=480&width=870&top_left_y=842&top_left_x=593)

We can write the derived proportions as a system of equations

$$
\begin{array}{r}
x + y - 3z = 0, \\
2x - 3y - 3z = 0.
\end{array}
$$

By subtracting these equations, we obtain the equality $x = 4y$ or $x : y = 4 : 1$. Substituting this result into the first equation, we get $5y = 3z$ or $y : z = 3 : 5$. By combining both ratios, we obtain the result $x : y : z = 12 : 3 : 5$.

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