jmc

Since a $13-14-15$ triangle is a $5-12-13$ triangle and a $9-12-15$ triangle "glued" together on the $12$ side, $[ABC]=\frac{1}{2}\cdot 12\cdot 14=84$. There are six points of intersection between $\Delta ABC$ and $\Delta A^{\prime }{B}^{\prime }{C}^{\prime }$. Connect each of these points to $G$. There are $12$ smaller congruent triangles which make up the desired area. Also, $\Delta ABC$ is made up of $9$ of such triangles. Therefore, $\left[\Delta ABC\bigcup \Delta A^{\prime }{B}^{\prime }{C}^{\prime }\right]=\frac{12}{9}[\Delta ABC]=\frac{4}{3}\cdot 84=\boxed{112}$.