In the provided text about triangle \( \Delta ABC \) and triangle \( \Delta A'B'C' \), the task involves proving certain geometric properties related to angles and lines. Here are the key points to consider based on the information:

### Given:
- Triangle \( \Delta ABC \) is a right triangle at \( A \).
- \( AB < A'B' \) and \( AC < A'C' \).
- \( AH \) is the height from \( A \) to side \( BC \).
- Points \( E, F, M \) are midpoints of segments \( AB \), \( AC \), and \( BC \) respectively.

### Tasks:
1. **Part a**: Show that the triangle \( AHCN \) is right-angled.
- To do this, you would leverage the properties of midpoints and right triangles. Use the fact that \( F \) is the midpoint of \( HN \) to demonstrate that angles \( AHF \) and \( HN \) satisfy the right triangle condition.

2. **Part b**: Prove that line segment \( ME \) is perpendicular to \( DE \).
- This involves using the properties of midpoints and right angles. You might consider the slopes of segments to demonstrate orthogonality.

3. **Part c**: Prove that \( HE \) is perpendicular to \( HF \).
- This could involve geometric reasoning or the use of coordinate geometry depending on the framework applied.

### General Approach for Proofs:
- Use properties of right triangles.
- Leverage the definition of midpoints.
- Consider geometric principles such as the Pythagorean theorem and slope conditions for perpendicular lines.

If you need more detailed strategies or specific theorems to employ for each part, let me know!