2. Given a triangle $A B C, \angle B=90^{\circ}$. On the sides $A C, B C$ points $E$ and $D$ are chosen respectively, such that $A E=E C, \angle A D B=\angle E D C$. Find the ratio $C D: B D$.
Answer: $2: 1$

Solution. Construct triangle $A^{\prime} B C$, symmetric to the given one with respect to side $B C$. Points $A^{\prime}, D, E$ lie on the same line, since $\angle A^{\prime} D B = \angle E D C$. Therefore, $D$ is the point of intersection of the medians $A^{\prime} E$ and $C B$ of triangle $A A^{\prime} C$, and divides them in the ratio $2: 1$ from the vertex.

![](https://cdn.mathpix.com/cropped/2024_05_06_cde1c5eea74bc8e8715ag-03.jpg?height=860&width=893&top_left_y=701&top_left_x=593)

Criteria.

(-.) A special case was considered where the correct answer was obtained, but in the solution, it was obtained that in the triangle, an angle is equal to 30. Or in the solution, similarity was obtained, where the correct answer was found when determining the coefficient.

(-/+) The correct answer was obtained, but it was not proven why $\mathrm{K}=2$ and there is no similarity in which this coefficient could be found.

(+/2) The correct answer was obtained, but the solution contains relationships that are used in the solution, but it is not proven why.

(+/-) The correct answer was obtained, and it is shown from which similarity the required similarity coefficient comes, but it is not explained why (i.e., it is clear from the solution why this is so, but there is no reference to this in the proof). Or the answer was not obtained, but from the solution, it is clear that the required coefficient can be obtained to derive the correct answer, but for some reason it was not done.

(+.) An arithmetic error in the correct solution led to an incorrect answer. Or the correct answer was obtained, but there are errors in notation in the solution, which make the solution unclear.