In Fig.9.23, E is any point on median AD of a ∆ ABC. Show that ar (ABE) = ar (ACE).

We know that the median divides a triangle into two triangles of equal triangle AD is the median for triangle ABC and ED is the median a triangle EBC. 

Since AD is the median of ΔABC. Therefore, it will divide ∆ABC into two triangles of equal areas.

Hence, Area (ΔABD) = Area (ΔACD)  ... (1)

Similarly, ED is the median of ΔEBC.

Hence, Area (ΔEBD) = Area (ΔECD)  ... (2)

Substract equation (2) from equation (1), we obtain

ar (ΔABD) - ar (ΔEBD) = ar (ΔACD) - ar (ΔECD)

Area (ΔABE) = Area (ΔACE)