If a ^ 2 + b ^ 2 + c ^ 2 - ab - bc - ca = 0 prove that a = b = c

We are given the equation:

\[
a^2 + b^2 + c^2 - ab - bc - ca = 0
\]

We need to prove that \(a = b = c\).

### Proof:

1. **Rewrite the equation**: 

   The given equation can be rearranged as:

   \[
   a^2 + b^2 + c^2 - ab - bc - ca = 0
   \]

   This can also be expressed as:

   \[
   \frac{1}{2} \left( 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca \right) = 0
   \]

2. **Use the identity**:

   Recognize that the expression inside the parentheses can be factored using a well-known identity:

   \[
   2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = (a - b)^2 + (b - c)^2 + (c - a)^2
   \]

   Therefore, the equation becomes:

   \[
   \frac{1}{2} \left( (a - b)^2 + (b - c)^2 + (c - a)^2 \right) = 0
   \]

3. **Interpret the equation**:

   Since the sum of squares is zero, each individual square must be zero because the square of a real number is non-negative:

   \[
   (a - b)^2 = 0, \quad (b - c)^2 = 0, \quad (c - a)^2 = 0
   \]

4. **Solve the system**:

   Solving each equation gives:

   \[
   a - b = 0 \quad \Rightarrow \quad a = b
   \]
   \[
   b - c = 0 \quad \Rightarrow \quad b = c
   \]
   \[
   c - a = 0 \quad \Rightarrow \quad c = a
   \]

   Therefore, we conclude:

   \[
   a = b = c
   \]

This completes the proof.

Would you like more details, or do you have any questions? 

Here are some related questions to explore:
1. How does this identity simplify when dealing with complex numbers?
2. Can this approach be extended to vectors in three-dimensional space?
3. What happens if we allow non-real numbers for \(a\), \(b\), and \(c\)?
4. How does this identity relate to geometric representations of equilateral triangles?
5. Can this result be generalized for higher dimensions with similar equations?

**Tip**: Always look for symmetry in equations, as it often leads to simplifications in proofs.

Understand how to prove that if a^2 + b^2 + c^2 - ab - bc - ca = 0, then a = b = c using symmetric polynomial identities and algebraic manipulation. Suitable for students in Grades 10-12.

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