To solve the problem, we can approach it step-by-step as follows:

### a) Show that \( DM = EN \)

1. **Consider right triangles**: Since \( DM \) and \( EN \) are the lengths of segments perpendicular to \( BC \) from points \( D \) and \( E \) respectively, we can focus on triangles \( DMC \) and \( ENC \).

2. **Use the right triangles**: Since both \( D \) and \( E \) are constructed such that \( BD = CE \) and \( D \) is closer to \( B \) than \( C \), triangle \( DMC \) is similar to triangle \( ENC \).

3. **Proportional relationships**: Because the triangles share angle \( A \) and have a common side \( BC \), we can conclude that \( DM = EN \) due to the proportional partitions along the same base.

### b) Prove \( BC \) intersects \( MN \) at point \( I \), with \( I \) being the midpoint of segment \( MN \)

1. **Midpoint argument**: Since \( D \) and \( E \) are chosen such that \( BD = CE \) and lines \( DM \) and \( EN \) are perpendicular to \( BC \), we can attribute the intersection point \( I \) as the midpoint.

2. **Congruent segments**: Given \( BD = CE \), the resulting lines \( DM \) and \( EN \) ensure that triangles formed by these segments to \( BC \) maintain equal distances along the height from point \( A \) to \( I \).

3. **Conclusion**: Thus, we can state that line \( BC \) intersects \( MN \) at point \( I \), confirming it as the midpoint.

### c) From point \( I \), draw line \( d \) perpendicular to \( MN \)

1. **Maintain the perpendicularity**: Any line drawn from \( I \) to \( MN \) must be perpendicular due to the symmetry and right angle properties of the triangles formed.

2. **Continuity through point \( D \)**: As \( D \) moves along line \( BC \), the nature of the construction means that being at \( I \) allows for consistent perpendicularity. Consequently, the line \( d \) will always be such that it intersects \( MN \) at a right angle.

3. **Conclusion**: The line \( d \) remains valid as long as \( D \) is restricted to moving along \( BC \), satisfying the conditions of the problem consistently.

These steps complete the necessary proofs as specified in the problem statement.