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Let's carefully analyze the question and the options given. The question asks: **Do any three points always, sometimes, or never determine a plane?**

### Mathematical Analysis:

1. **Plane Determination by Points:**
   - **Three points** can determine a plane, but it depends on their arrangement.
   - If the points are not collinear (i.e., they do not all lie on the same straight line), then **they always determine a unique plane**.
   - If the points are collinear, they do not determine a unique plane, since an infinite number of planes can pass through a single line.

### Evaluating the Options:

- **Option 1: "never; It takes at least four points to determine a plane."**
  - This is incorrect. Three non-collinear points are sufficient to determine a plane.

- **Option 2: "sometimes; The three points may be collinear."**
  - This is partially correct. If the points are collinear, they do not determine a unique plane. However, this option does not fully address the situation when the points are not collinear, in which case they do determine a unique plane.

- **Option 3: "sometimes; The three points may not lie in the same plane."**
  - This is incorrect. In 3-dimensional space, any set of three points will always lie in a plane (unless we are considering more complex geometries such as higher dimensions).

- **Option 4: "always; Any three points lie in a plane."**
  - This is mostly correct, as three points always lie in a plane. However, it does not specify that the plane may not be unique if the points are collinear.

- **Option 5: "always; There is exactly one plane through any three points."**
  - This is the correct answer. Three points **always** determine a plane, and if the points are non-collinear, there is exactly one unique plane through those points.

### Conclusion:
The correct answer is: **"always; There is exactly one plane through any three points."**

Would you like more details on this topic, or do you have any other questions? Here are some related questions to deepen your understanding:

1. What are the conditions for four points to determine a plane?
2. Can a single point determine a plane? Why or why not?
3. How does the concept of collinearity affect the determination of a plane?
4. What is the minimum number of points required to determine a line?
5. How do different dimensions (e.g., 2D, 3D) affect the concept of determining planes?

**Tip:** Visualizing the relationship between points and planes in 3D space can help solidify your understanding of these geometric principles.

Do any three points always, sometimes, or never determine a plane?

Explore the geometric principles of plane determination by three points, and learn when points are sufficient to define a unique plane. This problem covers key concepts like collinearity and plane geometry, suitable for students in Grades 8-10.

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