Question

If the centroid of ∆ABC, which has vertices A(a, b), B(b, c) and C(c, a), is the origin, find the value of (a + b + c).

Answer 1

The given points are A(a, b), B(b, c) and C(c, a).
Here,
$\left(x_1=a,y_1=b\right),\left(x_2=b,y_2=c\right)\mathrm{and}\left(x_3=c,y_3=a\right)$
Let the centroid be (x, y).
Then,
$$\begin{gathered} x=\frac{1}{3}\left(x_1+x_2+x_3\right) \\ =\frac{1}{3}\left(a+b+c\right) \\ =\frac{a+b+c}{3} \\ y=\frac{1}{3}\left(y_1+y_2+y_3\right) \\ =\frac{1}{3}\left(b+c+a\right) \\ =\frac{a+b+c}{3} \end{gathered}$$
But it is given that the centroid of the triangle is the origin.
Then, we have:
$$\begin{gathered} \frac{a+b+c}{3}=0 \\ \Rightarrow a+b+c=0 \end{gathered}$$