If $\alpha, \beta, \gamma$ be the roots of the equation $2x^3 + 3x^2 - 12x + 3 = 0$ and $A (\alpha, \beta, \gamma), B (\beta, \gamma, \alpha) C(\gamma, \alpha, \beta)$ represent vertices of a triangle ABC then the centroid of the triangle lies upon the line

Using the centroid formula, the x coordinate is (alpha + beta + gamma)/3, the y coordinate is (beta + gamma + alpha)/3, and the z coordinate is (gamma + alpha + beta)/3. Since the numerators are identical, the coordinates are equal, placing the centroid on the line x = y = z.

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