The medians `B E` and `C F` of a triangle `A B C` intersect at `G` . Prove that area of ` G B C=a r e aofq u a d r i l a t e r a lA G F Edot`

Explanation

Given: The medians `BE` and `CF` of a triangle `ABC` intersect at `G`
To prove: `ar(ΔGBC)=ar(AFGE)`
Proof: `BE` is the median of `ΔABC`
`⇒ar(ΔBEC)=1/2ar(ΔABC)` __(1)
Median of triangle divides into two triangles of equal area.
Also `CF` is median of `ΔABC`
`⇒ar(ΔACF)=1/2ar(ΔABC)` __(2)
From equation (1) and equation (2), we get:
`ar(ΔACF)=ar(ΔBEC)`
`ar(ΔGBC)+ar(ΔGEC)=ar(AFGE)+ar(GEC)`
`∴ar(ΔGBC)=ar(AFGE)`