8.3. In triangle $A B C$, point $D$ is the midpoint of side $A C$, and $E$ is an interior point of side $B C$ such that angles $B E A$ and $C E D$ are congruent. Find the numerical value of the ratio $\frac{A E}{D E}$.
Solution. On the extension of the ray $(ED$, we take the point $F$ such that $DE = DF$ (see figure).

Since $AD = DC$, it follows that the quadrilateral $AECF$ is a parallelogram. Then the lines $AE$ and $CF$ are parallel, as are the lines $AF$ and $BC$. Then we obtain

$$
m(\angle BEA) = m(\angle EAF) = \alpha = m(\angle CEF) = m(\angle AFE)
$$

It follows that triangle $AEF$ is isosceles with $AE = FE$. Since

$EF = 2 \cdot DE$, we have that $AE = 2 \cdot DE \Leftrightarrow \frac{AE}{DE} = 2$.

![](https://cdn.mathpix.com/cropped/2024_06_06_bf69806fd13b3a2e3925g-2.jpg?height=515&width=550&top_left_y=355&top_left_x=1362)