## Understanding the Formula (a - b)² + (b - c)² + (c - a)²

The formula **(a - b)² + (b - c)² + (c - a)²** is a useful algebraic expression that has applications in various mathematical and geometric contexts. It is used to represent the sum of squared differences between three variables, 'a', 'b', and 'c'.

### Expanding the Formula

The formula can be expanded by applying the algebraic identity: **(x - y)² = x² - 2xy + y²**.

Expanding the formula, we get:

- (a - b)² = a² - 2ab + b²
- (b - c)² = b² - 2bc + c²
- (c - a)² = c² - 2ca + a²

Adding all these together, we get:

**(a - b)² + (b - c)² + (c - a)² = 2a² + 2b² + 2c² - 2ab - 2bc - 2ca**

### Applications

This formula has various applications in different fields, including:

**1. Geometry:**

**Triangle Inequality:**The formula can be used to prove the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.**Distance Formula:**The formula relates to the distance formula, which calculates the distance between two points in a coordinate plane.

**2. Algebra:**

**Simplification of expressions:**The formula can be used to simplify complex algebraic expressions involving squared differences.**Solving equations:**The formula can be helpful in solving equations involving squared differences.

**3. Statistics:**

**Variance Calculation:**In statistics, the formula can be used to calculate the variance of a set of data points.

### Example

**Question:** Simplify the expression (2 - 3)² + (3 - 1)² + (1 - 2)²

**Solution:**

Using the formula:

- (2 - 3)² + (3 - 1)² + (1 - 2)² = 2(2²) + 2(3²) + 2(1²) - 2(2)(3) - 2(3)(1) - 2(1)(2)
- = 8 + 18 + 2 - 12 - 6 - 4
- = 6

Therefore, the simplified value of the expression is **6**.

### Conclusion

The formula (a - b)² + (b - c)² + (c - a)² is a fundamental algebraic expression with diverse applications in various mathematical and geometric contexts. Understanding its expansion and applications can be beneficial in solving a wide range of problems related to algebra, geometry, and statistics.