d. Find the midpoint of $$ A C $$ and compare this point with the point of intersection of the diagonals $$ A C $$ and $$ B D $$.

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To address this problem, let's analyze the relationships between the midpoint of diagonal $$ AC $$ and the point of intersection of the diagonals $$ AC $$ and $$ BD $$ in a quadrilateral $$ ABCD $$.

### 1. **Finding the Midpoint of $$ AC $$**

Assume the quadrilateral $$ ABCD $$ has vertices at coordinates:
- $$ A(x_1, y_1) $$
- $$ C(x_3, y_3) $$

The midpoint $$ M $$ of diagonal $$ AC $$ is calculated using the midpoint formula:
$$
M = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right)
$$

### 2. **Finding the Point of Intersection of Diagonals $$ AC $$ and $$ BD $$**

Let the other two vertices be:
- $$ B(x_2, y_2) $$
- $$ D(x_4, y_4) $$

To find the intersection point $$ O $$ of diagonals $$ AC $$ and $$ BD $$, set up the equations of the lines representing each diagonal and solve for their intersection.

For diagonal $$ AC $$:
$$
\text{Slope of } AC = m_{AC} = \frac{y_3 - y_1}{x_3 - x_1}
$$
$$
\text{Equation of } AC: y - y_1 = m_{AC}(x - x_1)
$$

For diagonal $$ BD $$:
$$
\text{Slope of } BD = m_{BD} = \frac{y_4 - y_2}{x_4 - x_2}
$$
$$
\text{Equation of } BD: y - y_2 = m_{BD}(x - x_2)
$$

By solving these two equations simultaneously, you find the coordinates of $$ O $$.

### 3. **Comparison of $$ M $$ and $$ O $$**

- **In a **_**Parallelogram**_**:**  
  In a parallelogram, the diagonals bisect each other. This means that the midpoint $$ M $$ of $$ AC $$ and the midpoint of $$ BD $$ coincide. Therefore, the point of intersection $$ O $$ of the diagonals $$ AC $$ and $$ BD $$ is exactly the midpoint $$ M $$ of $$ AC $$.

- **In a **_**General Quadrilateral**_**:**  
  In a general quadrilateral that is **not** a parallelogram, the diagonals do **not** necessarily bisect each other. Consequently, the midpoint $$ M $$ of $$ AC $$ will **not** coincide with the point of intersection $$ O $$ of the diagonals $$ AC $$ and $$ BD $$.

### **Illustrative Example**

**Parallelogram Example:**
Consider a parallelogram with vertices:
- $$ A(0, 0) $$
- $$ B(4, 0) $$
- $$ C(4, 3) $$
- $$ D(0, 3) $$

- Midpoint of $$ AC $$:
  $$
  M = \left( \frac{0 + 4}{2}, \frac{0 + 3}{2} \right) = (2, 1.5)
  $$
  
- Intersection point $$ O $$ of diagonals $$ AC $$ and $$ BD $$:
  Diagonals intersect at $$ (2, 1.5) $$, which is the same as $$ M $$.

**General Quadrilateral Example:**
Consider a quadrilateral with vertices:
- $$ A(0, 0) $$
- $$ B(4, 0) $$
- $$ C(2, 3) $$
- $$ D(1, 2) $$

- Midpoint of $$ AC $$:
  $$
  M = \left( \frac{0 + 2}{2}, \frac{0 + 3}{2} \right) = (1, 1.5)
  $$
  
- Intersection point $$ O $$ of diagonals $$ AC $$ and $$ BD $$:
  Solving the equations of $$ AC $$ and $$ BD $$, suppose they intersect at $$ \left(\frac{4}{3}, 2\right) $$.
  
- Here, $$ M = (1, 1.5) $$ does **not** coincide with $$ O = \left(\frac{4}{3}, 2\right) $$.

### **Conclusion**

The midpoint of diagonal $$ AC $$ coincides with the point of intersection of diagonals $$ AC $$ and $$ BD $$ **only if** the quadrilateral $$ ABCD $$ is a parallelogram. In general quadrilaterals, these two points do not necessarily coincide.
In a parallelogram, the midpoint of diagonal $$ AC $$ is the same as the point where diagonals $$ AC $$ and $$ BD $$ intersect. In other quadrilaterals, these points do not coincide.

d. Find the midpoint of and compare this point with the point of intersection of the diagonals and .

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