Cho ΔABC vuông tại A (AB < AC). Gọi M là trung điểm BC, trên tia AM lấy điểm D sao cho AM = MD (D ≠ A)

Let's break down the problem step-by-step based on the requirements:

### a) Chứng minh tứ giác ABCD là hình chữ nhật.

1. **Given:** Triangle \( ABC \) is right-angled at \( A \) with \( AB < AC \).
2. **Properties of Points:**
- \( M \) is the midpoint of \( BC \).
- \( D \) is on ray \( AM \), where \( AM = MD \).
3. **To Prove:**
- \( ABCD \) is a rectangle.
4. **Steps:**
- Since \( M \) is the midpoint of \( BC \), the coordinates of \( M \) can be used to establish symmetry.
- Also, since \( AM = MD \) and \( D \) lies along the line through \( A \) and \( M \), angles \( MAD \) and \( DAC \) can be analyzed.
- Using the right angles at \( A \) and the properties of isosceles triangles, you can conclude that opposite sides are equal and angles are right angles.

### b) Chứng minh rằng tứ giác ADCN là hình bình hành.

1. **Given:** Point \( N \) on line \( BA \) such that \( BA = AN \).
2. **To Prove:** Quadrilateral \( ADCN \) is a parallelogram.
3. **Steps:**
- Since \( BA = AN \), it follows that \( AC \parallel DN \) and \( AD \parallel CN \).
- Hence, opposite sides are equal, confirming that \( ADCN \) is a parallelogram.

### c) Chứng minh ba điểm B, F, I thẳng hàng.

1. **To Prove:** Line \( AC \) intersects line \( CD \) at point \( I \) such that \( CI = CD \).
2. **Steps:**
- Using the properties of angles and triangles, particularly isosceles triangles formed by point \( F \) as midpoint, you can verify that \( B, F, I \) are collinear.
- Also, since \( F \) is also defined in relation to triangle \( AC \), this helps establish the collinearity through coordinate geometry or angle correspondence.

### d) Chứng minh rằng DK ⊥ KF.

1. **Given:** Points \( E, K \) defined such that \( K \) is the midpoint of line segment \( EB \).
2. **To Prove:** Line \( DK \) is perpendicular to line segment \( KF \).
3. **Steps:**
- You can use the midpoint theorem or properties of slopes involving right triangles.
- By establishing that the angle between lines \( DK \) and \( KF \) equals \( 90^\circ \), the proof can be formally concluded.

### Conclusion:

Delve deeper into each step, using geometric properties, theorems, and coordinate output where necessary to confirm the properties outlined above.