If you understand how you can use vectors in geometry, you'll find a vector proof just here.
Originally Posted by snigdha
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 areaSo, if we subtract the common area - the quadrilateral (yellow area + green area) - from each triangle, we get:In colours:
red area = blue areaBut we can also see that, because (red) and (yellow) have equal bases and the same height:In colours:
red area = yellow areaSimilarly:In colours:
green area = blue areaThus 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!)