Remember: Median divides a triangle into two triangles of equal area.
ar(ΔABD)=ar(ΔACD)....(1) (Given)
Therefore, AD is the median of ΔABC on base BC.
O is any point on AD. So, OD is the median of ΔOBC.
∴ ar(ΔOBD)=ar(ΔOCD).....(2)
Subtracting (2) from (1), we get
ar(ΔABD)-ar(ΔOBD)=ar(ΔACD)-ar(ΔOCD)
⇒ar(ΔABO)=ar(ΔACO)
Hence, statement (1) is correct.
G is the point of concurrence of the medians of ΔABC.
ar(ΔABG)=ar(ΔACG)....(3)
(Proof same as given above)
Also,ar(ΔBCG)=ar(ΔACG)...(4)
From (3) and (4), we have
ar(ΔABG)=ar(ΔBCG)=ar(ΔACG)
Hence, statement (2) is correct.