ABC and ABD are two Triangles on lthe same base AB. If line segment CD is bisected by AB at O, Show that ar(ABC) = ( ABD)

I know this....,

ΔABC and Δ ADC are on the same base AB (and on the opposite sides of AB....)

In ΔDAC, AO is the median, (given, CD is bisected by AB)

ar(DAO) = ar(CAO)  -----------{1} ( a median of a triangle divides it into 2 triangles of equal areas)

Similarlly, BO is the median of ΔCBD

∴ ar(DBO) = ar(CBO)  -----------{2}  ( a median of a triangle divides it into 2 triangles of equal areas)

Adding {1} and {2}

ar(DAO) + ar(DBO) = ar(CAO) + ar(CBO)

i.e., ar(ABD) = ar(ABC)

or,  ar(ABC) = ar(ABD)

 HENCE PROVED

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