M and N are mid-point of the diagnols AC and BD respectivley of quadrilateral ABCD, then`bar(AB) + bar(AD)+bar(CB)+bar(CD)`=

Answer

To solve the problem, we need to find the expression for the sum of the vectors \( \bar{AB} + \bar{AD} + \bar{CB} + \bar{CD} \) given that \( M \) and \( N \) are the midpoints of the diagonals \( AC \) and \( BD \) respectively in quadrilateral \( ABCD \).

### Step-by-Step Solution:

1. **Define the Position Vectors**:
Let the position vectors of points \( A, B, C, D \) be represented as:
\[ \vec{A} = \vec{a}, \quad \vec{B} = \vec{b}, \quad \vec{C} = \vec{c}, \quad \vec{D} = \vec{d} \]

2. **Find the Vectors**:
We need to express the vectors \( \bar{AB}, \bar{AD}, \bar{CB}, \bar{CD} \):
- \( \bar{AB} = \vec{B} - \vec{A} = \vec{b} - \vec{a} \)
- \( \bar{AD} = \vec{D} - \vec{A} = \vec{d} - \vec{a} \)
- \( \bar{CB} = \vec{B} - \vec{C} = \vec{b} - \vec{c} \)
- \( \bar{CD} = \vec{D} - \vec{C} = \vec{d} - \vec{c} \)

3. **Sum the Vectors**:
Now, we sum these vectors:
\[ \bar{AB} + \bar{AD} + \bar{CB} + \bar{CD} = (\vec{b} - \vec{a}) + (\vec{d} - \vec{a}) + (\vec{b} - \vec{c}) + (\vec{d} - \vec{c}) \]

4. **Combine Like Terms**:
Combining the terms:
\[ = (\vec{b} + \vec{d} - 2\vec{a}) + (\vec{b} + \vec{d} - 2\vec{c}) \]
\[ = 2\vec{b} + 2\vec{d} - 2\vec{a} - 2\vec{c} \]
\[ = 2(\vec{b} + \vec{d} - \vec{a} - \vec{c}) \]

5. **Midpoints**:
Since \( M \) is the midpoint of \( AC \):
\[ \vec{M} = \frac{\vec{a} + \vec{c}}{2} \]
And \( N \) is the midpoint of \( BD \):
\[ \vec{N} = \frac{\vec{b} + \vec{d}}{2} \]

6. **Substitute Midpoints**:
Now substituting \( \vec{M} \) and \( \vec{N} \) into our expression:
\[ \vec{b} + \vec{d} = 2\vec{N} \]
\[ \vec{a} + \vec{c} = 2\vec{M} \]

7. **Final Expression**:
Substitute these into the equation:
\[ \bar{AB} + \bar{AD} + \bar{CB} + \bar{CD} = 2\left(2\vec{N} - 2\vec{M}\right) = 4(\vec{N} - \vec{M}) \]

Thus, the final result is:
\[ \bar{AB} + \bar{AD} + \bar{CB} + \bar{CD} = 4(\vec{N} - \vec{M}) \]