Task B-2.3. (20 points) In a right triangle $A B C$, point $D$ is the foot of the altitude dropped from vertex $C$ to the hypotenuse $\overline{A B}$, point $E$ is the midpoint of segment $\overline{C D}$, and point $F$ is the intersection of lines $A E$ and $B C$. If $|A D|=4$ and $|B D|=9$, determine the length of segment $\overline{A F}$.
Solution. From the equality $|CD|^2 = |AD| \cdot |BD|$ for the right triangle $ABC$, we get $|CD| = 6$. Since $E$ is the midpoint of segment $\overline{CD}$, we have $|DE| = 3$. From the right triangle $AED$, we have $|AE| = 5$.

![](https://cdn.mathpix.com/cropped/2024_05_30_978576e245d98ac92b08g-11.jpg?height=409&width=524&top_left_y=1107&top_left_x=651)

Let $G$ be a point on side $\overline{BC}$ such that $DG \parallel AF$. Since $\overline{EF}$ is the midline of triangle $CDG$, with $|EF| = x$, we have $|DG| = 2x$.

From the similarity of triangles $AFB$ and $DGB$, we get the equality $\frac{|AF|}{|DG|} = \frac{|AB|}{|DB|}$, i.e., $\frac{5 + x}{2x} = \frac{13}{9}$, from which we find $x = \frac{45}{17}$. Finally, $|AF| = \frac{130}{17}$.