Cho ∆ABC vuông tại A có AB < AC, dựng điểm D nằm giữa AB và trung tuyến AE. Gọi E, F lần lượt là điểm trên AB, AC

To solve the problem with triangle \( \Delta ABC \) that is right-angled at \( A \) (where \( AB < AC \)), we'll analyze the statements one by one based on the geometric relationships.

### a) Prove that quadrilateral \( BDHE \) is a parallelogram.

1. **Coordinates and Properties**:
Let \( A(0, 0) \), \( B(b, 0) \), and \( C(0, c) \). The coordinates of points can help visualize the triangle \( ABC \).

2. **Midpoints**:
- Let \( D \) be the point on \( AB \) (fixed but can be chosen). The segments \( BD \) and \( DE \) need to be shown parallel.
- \( E \) is on \( AC \) and it also needs properties of a parallelogram.

3. **Vectors**:
Show that \( \overrightarrow{BD} \parallel \overrightarrow{EF} \) and \( \overrightarrow{BE} \parallel \overrightarrow{DF} \).

4. **Conclusion**:
If both vector pairs are parallel, \( BDHE \) is a parallelogram.

### b) Proving that quadrilateral \( DFHE \) is a trapezoid.

1. **Identify Bases**:
Since \( AB \) and \( AC \) are the two segments, \( DF \parallel HE \).

2. **Angle properties**:
Observe the angles formed by the transversal line \( AE \), establishing that the alternate interior angles are equal.

3. **Conclusion**:
Since one pair of opposing sides are parallel, \( DFHE \) is a trapezoid.

### c) Proving that \( A, N, M \) are collinear.

1. **Define Midpoints**:
Let \( M \) be the midpoint of \( EM \) and \( N \) be the midpoint of \( BN \).

2. **Show parallel lines**:
Use the properties of midpoints to establish ratios:
\[
AN : NM : MB
\]
Show that \( A, N, M \) can be represented through collinearity equations.

3. **Conclusion**:
Use the concept of slope or vector analysis to prove they are on the same line.

### Summary
Make sure to utilize properties of triangles, parallelograms, and trapezoids along with coordinate geometry to rigorously prove these geometric statements step by step. Each statement should draw on fundamental theorems about transversals, alternate angles, and midpoints.