The vertices of $\displaystyle \Delta \mathrm{ABC}$ are $\displaystyle \mathrm{A}(1,1,0), \mathrm{B}(1,2,1)$ and $\displaystyle \mathrm{C}(-2,2,-1)$. Find the equations of the medians through A and B . Use the equations so obtained to find the coordinates of the centroid.

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 ( $\displaystyle 0, \frac{5}{3}, 0$ )