In a triangle A B C , let G denote its centroid and let M, N be points in the interiors of the segments A B, A C , respectively, such that M , G , N are collinear. If r denotes the ratio of the area of triangle AMN to the area of ABC then

Step-by-step solution

Let A B = b , A C = c array l A M = b A N =m c array Let G divides MN in the ratio K : 1 So k c + b k+1 = b + c 3 array l k k +1 = 1 3 k +1 = 1 3 k = 1 + 1 =3 AM GM 1 + 1 = 1 2 2 ( 2 3 )² ...(1) array Now, area of AMN area of ABC = 1 2 | b c | 1 2 | b c | = using 1 + 1 =3 Ratio = 3 -1 [0,1] maximum value of ratio = ² 3 -1 attain when =1 using derivative but is not 1 becuase M is an interior point. so 4 9 ratio < 1 2