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The medians $A D, B E$, and $C F$ of triangle $A B C$ intersect at the centroid $G$. The line through $G$ that is parallel to $B C$ intersects $A B$ and $A C$ at $M$ and $N$, respectively. If the area of triangle $A B C$ is 810 , then find the area of triangle $A M N$.
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We know that $A G: A D=2: 3$. Triangles $A M G$ and $A B D$ are similar, so $A M: A B=A G: A D=2: 3$. Likewise, $A N: A C=A G: A D=2: 3$. Therefore, the area of triangle $A M N$ is $810 \cdot(2 / 3)^{2}=360$. Final Answer: The final answer is 360
by Diamond (66,897 points)

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