Triangles and areas
In a triangle , let denote its centroid and let , be points in the interiors of the segments , ,respectively, such that , , are collinear. If denotes the ratio of the area of triangle to the area of and (a and b are co-prime natural numbers.). Find the value of
NOTE:- This question was asked in KVPY -2013
The answer is 13.
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Area of PQR represented as [PQR].
Let line XGY || BC. Let D, E, F be midpoints of the sides as in the Fig.
AG/AD=2/3, so [AXY]/[ABC]=(2/3) * (2/3) =4/9..............(1)
Let M be between F and X. Then
[AMN]=[AXY] - [MGX] + [YGN]...........(2)
In triangle MGX and YGN, XG=GY, angle MGN= angle YGN.
It can be easily seen that MG < GN.
The product of two sides, and Sin of included angle in a triangle gives its area.
So [MGX] < [YGN].
So from (2), we get [AMN]>[AXY] for all M between F and X.
Similarly, when M is between B and X, N is between E and Y, we again get [AMN]>[AXY].
For M, G, N to be co-linear, M ca n be only between F and B.
So [AXY]=[AMN] is the minimum area. From (1) a/b=4/9. So a+b=4+9= 1 3 .