Hello snigdha Originally Posted by
snigdha In a triangle ABC, BD and CE are the medians of a triangle, meet at centroid G.
Prove that BG=2GD.
If you understand how you can use vectors in geometry, you'll find a vector proof just here.
But if you want a more 'traditional' proof, look at the diagram I've attached.
In the diagram, and are the mid-points of and respectively.
If we consider the area of when its base is , we see that: area of area of because the base ( ) of is one-half of the base ( ) of , and the height of each triangle is the same.
Similarly, when we consider as the base of , we get: Therefore:With the colours I've used in the diagram, this is:blue area + green area + yellow area = red area + yellow area + green area
So, if we subtract the common area - the quadrilateral (yellow area + green area) - from each triangle, we get:In colours:red area = blue area
But we can also see that, because (red) and (yellow) have equal bases and the same height:In colours:red area = yellow area
Similarly:In colours:green area = blue area
Thus all four coloured triangles have the same area. ThereforeBut these triangles have a common base . Therefore the height of of the height of . So, by similar triangles:Tricky, isn't it? (It's much easier to use the vector method!)
Grandad