How can I prove mathematically that $(u+v+w)/3$ represents the centroid of the triangle formed by these vectors?

Question

Given $u ,v, w$ vector in different directions.

I can do it using a cad software.

Answer 1

Let $C=\frac{u+v+w}3$. The position vector of the mid-point of side $vw$ is $m=\frac{v+w}2$. It is easy to see that $u,m,C$ are colinear: $C-u\|m-u$. Thus $C$ lies on the median from $u$ to side $vw$. In a similar fashion you can show $C$ lies on the other medians.