If the centriod of a triangle formed by points $(a, b), (b, c)$ and $(c, a)$ is $(0, 0)$ , then find $a^{3} + b^{3} + c^{3}$ .

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Solution

$\begin{matrix} c e n t r o i d o f t r i a n g l e i s \Rightarrow {\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}} \Rightarrow \frac{a + b + c}{3} = 0 \Rightarrow a + b + c = 0 T h e n, a^{3} + b^{3} + c^{3} = 3 a b c A n s . \end{matrix}$

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a^{3} {(b - c)}^{3} + b^{3} {(c - a)}^{3} + c^{3} {(a - b)}^{3}

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