(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?
1. **Define the problem and given conditions:**
   We are given two triangles \( \triangle ABC \) and \( \triangle A'B'C' \). We denote by \( \alpha \) the angle between the altitude and the median from vertex \( A \) of triangle \( \triangle ABC \). Similarly, we define angles \( \beta \) and \( \gamma \) for vertices \( B \) and \( C \) of triangle \( \triangle ABC \), and angles \( \alpha' \), \( \beta' \), and \( \gamma' \) for vertices \( A' \), \( B' \), and \( C' \) of triangle \( \triangle A'B'C' \). It is given that \( \alpha = \alpha' \), \( \beta = \beta' \), and \( \gamma = \gamma' \).

2. **Restate the goal:**
   We need to determine if the given conditions imply that the triangles \( \triangle ABC \) and \( \triangle A'B'C' \) are similar.

3. **Analyze the given conditions:**
   The angles \( \alpha \), \( \beta \), and \( \gamma \) are the angles between the altitudes and the medians from the respective vertices. These angles are equal to the corresponding angles \( \alpha' \), \( \beta' \), and \( \gamma' \) in triangle \( \triangle A'B'C' \).

4. **Construct a counterexample:**
   Consider any triangle \( \triangle ABC \). Construct another triangle \( \triangle A'B'C' \) such that its sides are the medians of triangle \( \triangle ABC \). This construction is possible and straightforward.

5. **Verify the conditions for the counterexample:**
   In this construction, the angles between the altitudes and the medians in \( \triangle ABC \) will be equal to the corresponding angles in \( \triangle A'B'C' \). This is because the medians of \( \triangle ABC \) become the sides of \( \triangle A'B'C' \), and the angles between the medians and the altitudes are preserved.

6. **Check similarity:**
   However, the triangles \( \triangle ABC \) and \( \triangle A'B'C' \) do not have to be similar. The sides of \( \triangle A'B'C' \) are the medians of \( \triangle ABC \), and the medians of a triangle do not necessarily form a triangle similar to the original triangle.

7. **Conclusion:**
   Therefore, the given conditions \( \alpha = \alpha' \), \( \beta = \beta' \), and \( \gamma = \gamma' \) do not guarantee that the triangles \( \triangle ABC \) and \( \triangle A'B'C' \) are similar.

\[
\boxed{\text{False}}
\]