How to find the area of a triangle using the midpoints of it's medians?

Question

Say the midpoints of the medians of a triangle are (1,2),(8,2)&(1,8). Find it's area. Thanks

Answer 1

Let the centroid of the original triangle be denoted by $G$ and its vertices be $A$, $B$, $C$. Let $D$, $E$, $F$ be the midpoints of the medians. Then $\vec {GA}=4\vec{GD}$ etc. Therefore triangles $ABC$ and $DEF$ are similar, and $ABC$ has side-lengths $4$ times those of $DEF$ and so $16$ times its area.