Centroid of a Triangle

The centroid of a triangle is the point of concurrency of its medians and divides each median in the ratio of 2 : 1.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of the triangle ABC. Let AD be the median bisecting its base BC. Then, using mid-point formula,

$$ D = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) $$

Now, the point G on AD, which divides it internally in the ratio 2 : 1, is the centroid. If (x, y) are the co-ordinates of G, then

$$ x = \frac{x_1 + x_2 + x_3}{3} $$

$$ y = \frac{y_1 + y_2 + y_3}{3} $$