3 min read Jun 17, 2024

Exploring the Equation (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0

This equation, though seemingly complex, holds interesting mathematical properties. It represents a symmetric expression, meaning the variables a, b, and c can be interchanged without affecting the equation's validity. Let's delve deeper into its features and applications.

Simplifying the Equation

We can simplify the equation by expanding the products:


x² - bx - cx + bc + x² - ax - cx + ac + x² - ax - bx + ab = 0

3x² - 2(a+b+c)x + (ab + ac + bc) = 0

This simplified form is a quadratic equation in terms of x.

Solving for x

We can now solve for x using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 3, b = -2(a+b+c), and c = (ab + ac + bc).

Geometric Interpretation

The equation has a geometric interpretation. It represents the circumcircle of a triangle, where a, b, and c are the coordinates of the vertices. The solutions for x represent the x-coordinates of the points where the circumcircle intersects the x-axis.


This equation has various applications in:

  • Geometry: Finding the circumcircle of a triangle.
  • Algebra: Solving for roots of a quadratic equation.
  • Coordinate Geometry: Determining points of intersection between a circle and the x-axis.


The equation (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0, though appearing complex, is a symmetric quadratic equation with interesting geometric and algebraic interpretations. It finds applications in various mathematical areas, demonstrating the interconnectedness of different branches of mathematics.

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