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  • The medians BE and CF of a triangle ABC intersect at G. Prove that the area of GBC = area of the quadrilateral AFGE.

The medians BE and CF of a triangle ABC intersect at G. Prove that the area of \triangle GBC = area of the quadrilateral AFGE.

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Solution.

Given that the medians BE and CF of a triangle ABC intersect at G 
We know that the median divides a triangle into two triangles of same area.

 

So, ar(\triangle BEC)=\frac{1}{2}ar(\triangle ABC)  …(1)
And, ar(\triangle ACF)=\frac{1}{2}ar(\triangle ABC) ….(2)
from (1) and (2)
ar(\triangle ACF)=ar(\triangle BEC)
\therefore ar(\triangle BGC)=ar(GFAE)
This can be written as:
Area of \triangle GBC = area of the quadrilateral AFGE.
Hence proved

 

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