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The text is a geometric problem regarding two parallel lines \(a\) and \(b\) in space. It asks which of the given statements are true.

The statements are:

1. There exists a plane that contains both lines \(a\) and \(b\).
2. There exists a line that intersects both lines \(a\) and \(b\).
3. There exists a point that belongs to both lines \(a\) and \(b\).

### Analysis:
1. **Statement 1 (True):**  
   For two parallel lines in space, there always exists a plane that contains both lines. This is one of the fundamental properties of parallelism in three-dimensional space. Therefore, statement I is true.

2. **Statement 2 (False):**  
   If lines \(a\) and \(b\) are parallel, by definition, they do not intersect. Therefore, there cannot be a third line that intersects both parallel lines. Thus, statement II is false.

3. **Statement 3 (False):**  
   Parallel lines never meet, so there is no point that belongs to both lines \(a\) and \(b\). Therefore, statement III is false.

### Conclusion:
The only true statement is **I**.

Would you like any further clarification on this? 

Here are 5 related questions to consider:
1. What defines parallelism in different geometric spaces (2D vs 3D)?
2. Can two skew lines in space share a plane?
3. How does the definition of parallel lines change in non-Euclidean geometry?
4. What is the significance of the plane containing two parallel lines?
5. How can parallelism be proven in a geometrical proof?

**Tip:** Parallel lines in three-dimensional space always lie in the same plane, unlike skew lines, which do not intersect and are not in the same plane.

У просторі задані дві паралельні прямі a і b. Які з тверджень вірні? I. Існує площина, яка містить дві прямі a і b. II. Існує пряма, яка перетинає прямі a і b. III. Існує точка, яка належить обом прямим a і b.

Explore the geometric properties of parallel lines a and b in 3D space. This problem covers key concepts such as the existence of a plane containing parallel lines, the impossibility of a common intersection, and the basic theorems of parallelism. Suitable for students in Grades 9-12.

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