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.
+421 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.
