Prove, by vector method, that the diagonals of a parallelogram bisect each other , conversely, if the diagonals of a quadrilateral bisect each other, it is a parallelogram.

Answer

To prove that the diagonals of a parallelogram bisect each other and conversely, if the diagonals of a quadrilateral bisect each other, it is a parallelogram, we can use vector algebra.

### Step-by-Step Solution:

**Step 1: Define the Points of the Parallelogram**
Let the vertices of the parallelogram be denoted as \( A, B, C, D \). The diagonals are \( AC \) and \( BD \).

**Step 2: Represent the Points as Vectors**
Let the position vectors of points \( A, B, C, \) and \( D \) be represented as \( \vec{a}, \vec{b}, \vec{c}, \) and \( \vec{d} \) respectively.

**Step 3: Use the Properties of a Parallelogram**
In a parallelogram, we have the property that:
\[ \vec{b} - \vec{a} = \vec{c} - \vec{d} \]
This implies that:
\[ \vec{b} + \vec{d} = \vec{a} + \vec{c} \]

**Step 4: Find the Midpoint of the Diagonals**
The midpoint \( O \) of diagonal \( AC \) can be expressed as:
\[ \vec{o} = \frac{\vec{a} + \vec{c}}{2} \]
Similarly, the midpoint \( O \) of diagonal \( BD \) can be expressed as:
\[ \vec{o} = \frac{\vec{b} + \vec{d}}{2} \]

**Step 5: Set the Midpoints Equal**
Since both expressions represent the same point \( O \), we equate them:
\[ \frac{\vec{a} + \vec{c}}{2} = \frac{\vec{b} + \vec{d}}{2} \]
Multiplying through by 2 gives:
\[ \vec{a} + \vec{c} = \vec{b} + \vec{d} \]
This shows that the diagonals bisect each other at point \( O \).

**Step 6: Prove the Converse**
Now, let’s assume that the diagonals \( AC \) and \( BD \) of a quadrilateral bisect each other at point \( O \). This means:
\[ \vec{a} + \vec{c} = \vec{b} + \vec{d} \]
From this, we can express:
\[ \vec{b} - \vec{a} = \vec{c} - \vec{d} \]
This implies that:
\[ \vec{b} + \vec{d} = \vec{a} + \vec{c} \]
Thus, by the properties of vectors, we can conclude that:
\[ \text{AB} \parallel \text{CD} \quad \text{and} \quad \text{BC} \parallel \text{AD} \]
Since both pairs of opposite sides are parallel, the quadrilateral \( ABCD \) is a parallelogram.

### Conclusion:
We have shown that the diagonals of a parallelogram bisect each other and conversely, if the diagonals of a quadrilateral bisect each other, it is a parallelogram.