11.3. 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 $B E A$ is equal to angle $C E D$. Find the ratio of the lengths of segments $A E$ and $D E$.
Solution. Let $\angle B E A=\angle D E C=\varphi$. On the extension of $A E$ beyond point $E$, choose a point $P$ such that $E D=E P$, and on the extension of $D E$ beyond point $E$,

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choose a point $K$ such that $A E=E K$. $\angle R E K=\angle P E Q=\varphi$, therefore, $R E$ and $E Q$ are the perpendicular bisectors of $A K$ and $D P$ ( $R, Q$ are their feet). Triangles $R K E$ and $E P Q$ are right and similar, so $K E: E P=$ $R K: Q P$ or $A E: E D=A R: D Q$. $D Q$ is the midline in triangle $A R C$, hence, $A E: E D=2: 1$.

Answer: $A E: E D=2: 1$.