4.6. THE CENTROID OF A TRIANGLE

To find the coordinates of the centroid of the triangle whose vertices are (x ₁ , y ₁ , z ₁ ), (x ₂ , y ₂ , z ₂ ) and (x ₃ , y ₃ , z ₃ ).

Let A( x ₁, y ₁, z ₁), B( x ₂, y ₂, z ₂), C( x ₃, y ₃, z ₃) denote the vertices of the triangle ABC.

If D denotes the mid-point of BC, then its coordinates are

If G( x, y, z) denotes the centroid of the triangle ABC, then G divides the median AD in the ratio 2 : 1. [From elementary geometry]

?

Similarly,

Hence the centroid is