Point $D$ is the midpoint of side $A C$ of triangle $A B C$. On side $B C$, a point $E$ is chosen such that $\angle B E A = \angle C E D$. Find the ratio $A E: D E$.
Through point $D$, draw a line parallel to side $B C$.

## Solution

We will draw the midline $D F$ of triangle $C A E$, parallel to side $C E$. Triangle $D E F$ is isosceles because

$\angle D F E = \angle B E F = \angle D E C = \angle F D E$. Therefore, $A E: D E = A E: F E = 2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_808033c11c6fe49d1738g-20.jpg?height=597&width=692&top_left_y=519&top_left_x=709)

## Answer

$2: 1$.

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