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This problem involves a parallelogram \( ABCD \), where points \( E \) and \( F \) divide the line segment \( BC \) into three equal parts, and points \( G \), \( H \), and \( I \) divide \( AD \) into four equal parts. The task is to find the ratio of the areas of triangles \( \triangle HBF \) and \( \triangle EGI \).

### Step-by-step approach:

1. **Understanding the geometry of the parallelogram**:
   - The parallelogram is divided into several smaller triangles by drawing lines from points \( E \), \( F \), \( G \), \( H \), and \( I \).
   - \( E \) and \( F \) divide \( BC \) into three equal segments.
   - \( G \), \( H \), and \( I \) divide \( AD \) into four equal segments.

2. **Using the properties of similar triangles**:
   - Triangles inside parallelograms formed by parallel lines and proportional segments will have areas proportional to the lengths of their bases and heights.

3. **Calculate the area ratio**:
   - \( \triangle HBF \) shares part of its area with other triangles formed within the parallelogram.
   - Similarly, \( \triangle EGI \) is part of another group of triangles formed by the same segmentation.
   - By analyzing the geometry and the segmentations, we can find the ratio of their areas.

4. **Conclusion**:
   Based on the symmetry and division properties of the parallelogram, the ratio of the areas of triangles \( \triangle HBF \) and \( \triangle EGI \) is \( 2:1 \).

So, the correct answer is **B) 2:1**.

Do you want further details on this explanation or have any questions? 

Here are 5 related questions you can explore:
1. How do you calculate the area of a triangle inside a parallelogram using its height and base?
2. What are the properties of triangles formed inside a parallelogram by segmenting sides?
3. How does the division of a parallelogram's sides into equal parts affect the area of the resulting triangles?
4. Can you explain the geometric principles behind similar triangles in a parallelogram?
5. How can the area of different triangles in a parallelogram be compared using proportions?

**Tip**: When solving geometry problems involving parallelograms, always pay attention to symmetry and how the sides are divided. This often helps simplify area comparisons!

如附圖，平行四邊形 ABCD 中，E、F 將 BC 三等分，G、H、I 將 AD 四等分，則△HBF 面積 : △EGI 面積 = ?

This problem involves finding the area ratio between two triangles within a parallelogram, where the sides are divided into equal segments. It uses the principles of similar triangles and the properties of parallelograms. Suitable for Grades 9-11.

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