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The medians AD, BE, and CF of triangle ABC intersect at the centroid G. The line through G that is parallel to BC intersects AB and AC at M and N , respectively. If the area of triangle AGM is 144, then find the area of triangle AGN.

 Nov 27, 2020
 #1
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Based on the information and because BD || GM, we have that ∆AMG ~ ∆ABD and ∆ANG ~ ∆ACD and ∆AMN ~ ∆ABC. 

 

By median ratios, 3AG = 2AD, so common ratio is 4:9 for all triangles' areas. So we have that GM = GN because they are both similar with the same ratio to BD=BC. Therefore, [AGN] = [AGM]. Thus, [AGN] = 144, and you can check this by finding area of AMN/2 and ABD then ABC... etc.

 Nov 27, 2020
 #2
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The median of a triangle divides the triangle into two triangles with equal areas. 

 

Line AG is the median of the triangle AMN, therefore the areas of triangles AMG and ANG are equal. 

 Nov 27, 2020

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