The vertices of Δ ABC are A (1,1,0), B (1,2,1) and C (-2,2,-1). Find the equations of the medians through A and B. Use the equations so obtained to…

The mid-point of the BC is ( $\displaystyle \frac{-1}{2}, 2,0$ ) The equation of the median through A is $$\begin{align*} & \frac{x-1}{\dfrac{-1}{2}-1}=\frac{y-1}{2-1}=\frac{z}{0} \\ & \Rightarrow \frac{x-1}{-3}=\frac{y-1}{2}=\frac{z}{0} \tag{1} \end{align*} $$ The mid-point of the AC is $\displaystyle \left(\frac{-1}{2}, \frac{3}{2}, \frac{-1}{2}\right)$ The equation of the median through B is $$\begin{align*} & \frac{x-1}{\dfrac{-1}{2}-1}=\frac{y-2}{\dfrac{3}{2}-2}=\frac{z-1}{\dfrac{-1}{2}-1} \\ & \Rightarrow \frac{x-1}{-3}=\frac{y-2}{-1}=\frac{z-1}{-3} \ldots \ldots \ldots(2) \tag{2} \end{align*} $$ Any point on the line ($\displaystyle 1$) is ( $\displaystyle -3 \lambda+1,2 \lambda+1,0$ ) Any point on the line ($\displaystyle 2$) is ( $\displaystyle -3 \mu+1,-\mu+2,-3 \mu+1$ ) For the point of intersection, $$\begin{aligned} & -3 \lambda+1=-3 \mu+1,2 \lambda+1=-\mu+2,0=-3 \mu+1 \\ & \Rightarrow \lambda=\mu=\frac{1}{3} \end{aligned} $$ The coordinates of the centroid are ( $\displaystyle 0, \frac{5}{3}, 0$ )