In the given figure, ΔABC and ΔDBC have the same base BC. If AD and BC intersect at O, prove that

Given: ∆ ABC and ∆ DBC have the same base BC.

AD and BC intersect at O.

To show:

First, we construct the altitudes, AE and DF, of ∆ ABC and ∆ DBC, respectively.

Consider, ∆ AOE and ∆ DOF,

∠DFO = ∠AEO [Right angles]

∠DOF = ∠AOE [Vertically Opposite angles]

So, by AA criterion,

∆AOE ∼ ∆DOF

⇒ Ratio of all the corresponding sides of ∆ AOE and ∆ DOF are equal.

……….(i)

Now, we know that

……….(ii)

Similarly, ……….(iii)

Dividing (ii) by (iii),

[From (i)]