HMMT-Nov 2012 gen, Problem 9

Consider triangle $A B C$ where $B C=7, C A=8$, and $A B=9 . D$ and $E$ are the midpoints of $B C$ and $C A$, respectively, and $A D$ and $B E$ meet at $G$. The reflection of $G$ across $D$ is $G^{\prime}$, and $G^{\prime} E$ meets $C G$ at $P$. Find the length $P G$.

Solution: Observe that since $G^{\prime}$ is a reflection and $G D=\frac{1}{2} A G$, we have $A G=G G^{\prime}$ and therefore, $P$ is the centroid of triangle $A C G^{\prime}$. Thus, extending $C G$ to hit $A B$ at $F, P G=\frac{1}{3} C G=$ $\frac{2}{9} C F=\frac{2}{9} \sqrt{\frac{2\left(8^{2}+7^{2}\right)-9^{2}}{4}}=\frac{\sqrt{145}}{9}$ by the formula for the length of a median. $\fbox{\frac{\sqrt{145}}{9}}$.

Answer: \frac{\sqrt{145}}{9}