Question

By vector method prove that the medians of a triangle are concurrent.

Answer 1

Consider ΔABC. 

Let P, Q, R be the midpoints of the sides BC, CA, AB respectively.

Let \(\overline{a}\), \(\overline{b}\), \(\overline{c}\), \(\overline{p}\), \(\overline{q}\), \(\overline{r}\), \(\overline{g}\) be the position vectors of the points A, B, C, P, Q, R, G respectively. 

Since P, Q, R are the mid-points of the sides BC, CA, AB respectively

∴ By midpoint formula, we get

This shows that the point G whose position vector is \(\overline{g}\) lies on the three medians AP, BQ, CR dividing them internally in the ratio 2:1. 

Hence, the three medians are concurrent.