Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

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 $P G$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

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.

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