In triangle ABC the centroid is G(4, 3) and D(6, 5) is the midpoint of BC.
G divides the median AD in the ratio 2 : 1 measured from A. Find the coordinates of vertex A.

Let $A=(x,y)$. Since $AG$ : GD = 2 : 1, $G=(\frac{2\cdot 6+x}{3},\frac{2\cdot 5+y}{3})=$ (4, 3). So $12+x=12$, giving $x=0$, and $10+y=9$, giving $y=-1$; hence $A$ = (0, -1). (3, 2) solves the same equations with the ratio taken as 1 : 2; (2, 1) treats $G$ as the MIDPOINT of AD instead of the two-thirds point; (8, 7) steps from D away from A instead of towards it.