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In the given problem, we have a parallelogram \( ABCD \), and points \( E \), \( F \), \( G \), and \( H \) are the midpoints of segments \( AB \), \( OC \), \( CD \), and \( OA \) respectively. We know the area of \( ABCD \) is 112, and we need to find the area of quadrilateral \( EFGH \).

### Solution

1. **Observe the Structure and Midpoints**:
   Since \( E \), \( F \), \( G \), and \( H \) are midpoints of the segments, we can apply a property related to the midpoints in a parallelogram. Specifically, in any parallelogram, the quadrilateral formed by joining the midpoints of the sides has an area equal to half the area of the original parallelogram.

2. **Calculate Area of \( EFGH \)**:
   Since \( EFGH \) is formed by the midpoints of the sides and diagonals of parallelogram \( ABCD \), the area of \( EFGH \) is:
   \[
   \text{Area of } EFGH = \frac{1}{2} \times \text{Area of } ABCD
   \]

3. **Substitute the Given Area**:
   Given that the area of \( ABCD \) is 112, we substitute:
   \[
   \text{Area of } EFGH = \frac{1}{2} \times 112 = 56
   \]

### Conclusion
The area of quadrilateral \( EFGH \) is **56**.

Would you like further details or have any questions?

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### Related Questions
1. What is the general formula for the area of a quadrilateral formed by midpoints in any parallelogram?
2. How would the area of \( EFGH \) change if \( ABCD \) were a different type of quadrilateral?
3. Can this midpoint-area property be applied to other polygons? If so, how?
4. What are other methods to verify the area of \( EFGH \) without using the midpoint property?
5. How does this midpoint theorem relate to vectors in geometry?

### Tip
When dealing with midpoint-related problems in geometry, remember that connecting midpoints in parallelograms or triangles often results in shapes whose areas are fractions of the original figure, simplifying calculations significantly.

In the figure above, ABCD is a parallelogram, E, F, G, H are the midpoints of AB, OC, CD, and OA respectively. If the area of ABCD is 112, find the area of EFGH.

This problem explores how to find the area of a quadrilateral formed by the midpoints of a parallelogram's sides and diagonals. Given the area of parallelogram ABCD is 112, learn how to calculate the area of quadrilateral EFGH using midpoint properties.

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