In triangle PQR with P(0, 6), Q(-4, -2) and R(10, 2), the median from P meets QR at M.
Find the point that divides PM in the ratio 2 : 1 from P, and say what that point is called.

$M$ = midpoint of QR = $(\frac{-4+10}{2},\frac{-2+2}{2})=$ (3, 0). Dividing PM from P(0, 6) in the ratio 2 : 1: $x=\frac{2\cdot 3+1\cdot 0}{3}=2$ and $y=\frac{2\cdot 0+1\cdot 6}{3}=2$, so the point is (2, 2) — the centroid, confirmed by $(\frac{0-4+10}{3},\frac{6-2+2}{3})=$ (2, 2). Choosing (3, 0) stops at $M$ and never divides PM; (1, 4) applies the ratio as 1 : 2 with m and n swapped; (1.5, 3) treats the centroid as the midpoint of the median, but it lies two-thirds of the way along.