Let ABC be any triangle whose vertices A,B and C respectively.
We consider that D,E and F be mid point of ΔABC and G be the centroid ΔABC
Then, AE,BF&CD are medians
To prove- area ΔAGB= area ΔBGC area ΔAGB= area ΔBGC= area ΔCGA=13 area ΔABC
Now,
In ΔAGB and ΔAGC
AG is the median
Therefore ,
arΔAGB=arΔAGC …….. (1)
Similarly,
BG is the median
Therefore,
arΔAGB=arΔBGC …… (2)
By equation (1)&(2) to,
arΔAGB=arΔAGC=arΔBGC ……. (3)
Now, We know that
arΔAGB+arΔBGC+arΔCGA=arΔABC ......(4)
By equation (3) and (4) to, we get
arΔAGB+arΔAGB+arΔAGB=arΔABC,
arΔAGB=13ar△ABC
Substituting in equation(1),
arΔAGB=arΔBGC=arΔCGA=13arΔABC
Hence proved.