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.