A point D is taken on the side BC of a ΔABC, such that BD = 2DC.

Given that,

In ΔABC, BD = 2DC

To prove: ar(ΔABD) = 2ar(ΔADC).

Construction:

Take a point E on BD such that BE = ED

Proof: Since, BE = ED and BD = 2 DC

Then, BE = ED = DC

We know that median of triangle divides it into two equal triangles.

∴ In ΔABD, AE is the median.

Then, ar(ΔABD) = 2ar(ΔAED) ⋅⋅⋅⋅ (1)

In ΔAEC, AD is the median.

Then, ar(ΔAED) = 2ar(ΔADC) ⋅⋅⋅ (2)

Compare equation 1 and 2

ar(ΔABD) = 2ar(ΔADC).