3. $A B C D$ — a convex quadrilateral, where $A B=7, B C=4, A D=D C, \angle A B D=\angle D B C$. Point E is on segment $A B$ such that $\angle D E B=90^{\circ}$. Find the length of segment $A E$. (Spain, local phase, 2020-2021)
Answer: 1.5. Solution: From point $B$, on ray $B A$, we lay off segment $B F = B C$. Triangles $D B F$ and $D B C$ are congruent by two sides and the included angle. Therefore, $D F = D C = D A$, which means $D E$ is the height drawn from the vertex of isosceles triangle $A D F$. Since it is also the median, we have $A E = A F / 2 = (A B - B F) / 2 = (7 - 4) / 2 = 1.5$.