Question

Prove by vector method that the medians of equilateral triangle are equal

Answer 1

Consider △ABC as given below.

AD,BE and CF are the medians.

Let O be the origin and let O⃗ A=a⃗ ,O⃗ B=b⃗ ,O⃗ C=c⃗ ,O⃗ D=d⃗ ,O⃗ E=e⃗ ,O⃗ F=f⃗  be the position vectors of points A,B,C,D,E,F respectively.

From the mid-point formula, we get,

d⃗ =b⃗ +c⃗ 2,e⃗ =a⃗ +c⃗ 2,f⃗ =a⃗ +b⃗ 2.

⇒2d⃗ =b⃗ +c⃗ ,2e⃗ =a⃗ +c⃗ ,2f⃗ =a⃗ +b⃗ .

⇒2d⃗ +a⃗ 3=a⃗ +b⃗ +c⃗ 3,2e⃗ +b⃗ 3=a⃗ +b⃗ +c⃗ 3,2f⃗ +c⃗ 3=a⃗ +b⃗ +c⃗ 3.

⇒2d⃗ +a⃗ 2+1=2e⃗ +b⃗ 2+1=2f⃗ +c⃗ 2+1=a⃗ +b⃗ +c⃗ 3.

But a⃗ +b⃗ +c⃗ 3 is the position vector of the centroid of △ABC.

⇒ The centroid divides the medians AD,BE and CF in the ratio 2:1.

The point represented by a⃗ +b⃗ +c⃗ 3 (the centroid) lies on the medians AD,BE and CF.

⇒ The medians of a triangle are concurrent.