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.