Question

Suppose points A, B and C are not collinear and have coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃). Let D be the midpoint of BC and suppose G divides the median AD in the ratio 2 : 1. Find the coordinates of G and deduce that the medians of 4ABC are concurrent.

Answer 1

The midpoint D of BC is

((x₂ + x₃)/2, (y₂ + y₃)/2)

The point G that divides the median AD in the ratio 2 : 1 is

This is symmetric in x₁, x₂, x₃ and in y₁, y₂, y₃. So G lies on the other medians BE and C F, and the medians are concurrent.