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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 \(A M N\).
in Mathematics by Diamond (66,897 points) | 76 views

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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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