Show that the centroid of the triangle with vertices A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is
((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3).
Let D be the midpoint of BC. Join A and D.
Then, the coordinates of D are
x2+x3/2, y2+y3/2, z2+z3/2.
Let G be the centroid of triangle ABC. Then, G lies on AD and divides it in the ratio 2:1
∴ The coordinates of G are (Using section formula)
(2.(x2+x3/2)+1.x1)/(2+1), (2.(y2+y3/2)+1.y1)/(2+1), (2.(z2+z3/2)+1.z1)/(2+1)
i.e., (x1+x2+x3/3, y1+y2+y3/3, z1+z2+z3/3)