2. Let $A B C$ be a triangle with orthocenter $H$; suppose that $A B=13, B C=14, C A=15$. Let $G_{A}$ be the centroid of triangle $H B C$, and define $G_{B}, G_{C}$ similarly. Determine the area of triangle $G_{A} G_{B} G_{C}$.
Answer: $28 / 3$ Let $D, E, F$ be the midpoints of $B C, C A$, and $A B$, respectively. Then $G_{A} G_{B} G_{C}$ is the $D E F$ about $H$ with a ratio of $\frac{2}{3}$, and $D E F$ is the dilation of $A B C$ about $H$ with a ratio of $-\frac{1}{2}$, so $G_{A} G_{B} G_{C}$ is the dilation of $A B C$ about $H$ with ratio $-\frac{1}{3}$. Thus $\left[G_{A} G_{B} G_{C}\right]=\frac{[A B C]}{9}$. By Heron's formula, the area of $A B C$ is $\sqrt{21 \cdot 8 \cdot 7 \cdot 6}=84$, so the area of $G_{A} G_{B} G_{C}$ is $[A B C] / 9=84 / 9=28 / 3$.