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Let's solve part (b) of the given problem.

### Problem Recap:
We are given three points \( A = (2, 3, 4) \), \( B = (-1, 3, 10) \), and \( C = (6, 1, 9) \) in \( \mathbb{R}^3 \). We need to find a point \( P \) in \( \mathbb{R}^3 \) such that the points \( A \), \( B \), \( C \), and \( P \) form a rhombus. Additionally, we need to explain the reasoning behind the choice of \( P \).

### Step-by-Step Solution:

1. **Understanding the Rhombus Condition:**
   A rhombus is a quadrilateral where all four sides are of equal length. Therefore, for points \( A \), \( B \), \( C \), and \( P \) to form a rhombus, the distances between these points must satisfy the following conditions:
   \[
   |AB| = |BC| = |CP| = |PA|
   \]

2. **Vectors Calculation:**
   First, let's calculate the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) to help us determine the appropriate point \( P \).

   \[
   \overrightarrow{AB} = B - A = (-1 - 2, 3 - 3, 10 - 4) = (-3, 0, 6)
   \]

   \[
   \overrightarrow{AC} = C - A = (6 - 2, 1 - 3, 9 - 4) = (4, -2, 5)
   \]

3. **Midpoint of the Diagonals:**
   The diagonals of a rhombus bisect each other. Therefore, the point \( P \) should lie such that \( \overrightarrow{AP} \) is the same as \( \overrightarrow{PB} \) or \( \overrightarrow{CP} = \overrightarrow{PB} \).

   Let's calculate the midpoint \( M \) of \( AC \). This midpoint would be the same as the midpoint of \( BP \) in the rhombus.

   \[
   M = \left(\frac{A_x + C_x}{2}, \frac{A_y + C_y}{2}, \frac{A_z + C_z}{2}\right)
   \]
   Substituting the coordinates of \( A \) and \( C \):

   \[
   M = \left(\frac{2 + 6}{2}, \frac{3 + 1}{2}, \frac{4 + 9}{2}\right) = (4, 2, 6.5)
   \]

4. **Finding Point \( P \):**
   For the point \( P \) to be such that \( A, B, C, P \) form a rhombus, \( P \) must lie such that the midpoint of \( BP \) is also \( M \).

   Since \( M \) is the midpoint:
   \[
   P = 2M - B
   \]
   Substituting \( M = (4, 2, 6.5) \) and \( B = (-1, 3, 10) \):

   \[
   P = (2 \times 4 - (-1), 2 \times 2 - 3, 2 \times 6.5 - 10)
   \]
   Simplifying, we get:

   \[
   P = (8 + 1, 4 - 3, 13 - 10) = (9, 1, 3)
   \]

### Final Answer:
The point \( P = (9, 1, 3) \) is such that \( A, B, C, P \) form a rhombus.

**Reasoning Recap:**
The solution is based on the fact that the diagonals of a rhombus bisect each other, meaning the midpoint of one diagonal is the same as the midpoint of the other diagonal. By finding this midpoint and using the midpoint formula, we determined the location of the fourth point \( P \).

Would you like to delve into the details further or have any other questions?

Here are five related questions to explore:
1. How would the problem change if we were asked to form a rectangle instead of a rhombus?
2. Can you verify that the distances between the points \( A, B, C, P \) are equal?
3. What is the area of the rhombus formed by points \( A, B, C, \) and \( P \)?
4. How can we generalize the method to find the fourth point if the other three points are not in a plane?
5. If the points were in 2D space instead of 3D, how would the approach differ?

**Tip:** When solving problems involving quadrilaterals in space, consider the properties of the specific shape (like symmetry and equal distances) to guide the calculation of unknown points.

Learn how to find point P in 3D space such that points A, B, C, and P form a rhombus. Detailed step-by-step solution using geometry and vector calculations. Suitable for high school students studying coordinate geometry and vectors.

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