Answer: In a triangle, the medians divide the triangle into six smaller triangles with equal area. This can be proven using the fact that the medians of a triangle are concurrent, meaning they all intersect at a single point called the centroid.
Let's assume that AD = 2BE, then the area of △ADG is equal to four times the area of △BEG. This can be expressed as follows:
Area(△ADG) = 4 * Area(△BEG)
Since GD is one of the medians, it must be equal to one-third of AD. So, we can write:
GD = AD/3
Since the area of △ADG is equal to four times the area of △BEG, we can write:
Area(△ADG) = 4 * Area(△BEG)
(2BE)^2/2 * GD / 2 = 4 * BE^2/2 * EG / 2
Expanding and simplifying the above equation gives us:
BE^2 * GD / 2 = 4 * BE^2/2 * EG / 2
And, finally, dividing both sides of the equation by BE^2/2, we get:
GD = 1/3 * AD
This result holds true regardless of the relative lengths of AD and BE. Hence, the conclusion that GD = 1/3 AD is always true for any triangle △ABC where AD and BE are medians.
Explanation: