Cho hình ABCD, D là điểm A, E thuộc AC sao cho BD = CE. Gọi M là trung điểm của BC.DE. Vẽ các hình bình hành BDEN và CMNK. a. CMN = K không hàng. b. MN có AC là Q, B là P, C.MB = A.QP đẳng

To approach the problem, we can break down the steps based on the specifications given:

### Problem Outline:
1. **Construct Quadrilateral ABCD** with point D as A and point E on line AC such that \( BD = CE \).
2. **Find M** as the midpoint of line segment \( BC \).
3. **Draw Parallelograms** \( BDEN \) and \( CMNK \).

### Steps:
#### a. Show that \( CMN \) and \( K \) are not collinear:

1. **By definition of midpoints and parallelograms**:
- \( M \) is the midpoint of \( BC \), making \( BM = MC \).
- Since \( BDEN \) and \( CMNK \) are parallelograms, opposite sides are equal and parallel.

2. To prove that \( CMN \) and \( K \) are not collinear, we can use properties of parallel lines:
- Since \( CM \) is parallel to \( KN \), and if \( M \) lies on \( KN \), it would imply collinearity. Thus, if you determine that \( K \) does not lie on line \( CM \), it proves they are not collinear.

#### b. Show that \( MN \) is parallel to \( AC \) and that \( C \cdot MB = A \cdot QP \):

1. **Establishing \( MN \parallel AC \)**:
- From the properties of the parallelogram \( CMNK \), since \( CM \parallel KN \) and both sides can be shown parallel to \( AC \) through congruent angles or slopes, we verify that \( MN \) and \( AC \) are indeed parallel.

2. **Relation \( C \cdot MB = A \cdot QP \)**:
- This part likely refers to areas or lengths relating to segments formed by points \( A, Q, P, C, M, B \).
- By using the properties of similar triangles or congruent segments established by the parallelograms and midpoints, analyze the ratios to show \( C \cdot MB \) is equal to \( A \cdot QP \).

### Conclusion:
- Following through these steps should allow you to demonstrate that \( CMN \) and \( K \) are not collinear and the relationship between the segments as required. Drawing diagrams might also help visualize these relationships better.