ABCD is a quadrilateral with AB = AD. If AE and AF are internal bisectors of ∠BAC and ∠DAC respectively, then prove that EF||BD

Question

Philoid · Accepted Answer

Given: ABCD is a quadrilateral with AB = AD. If AE and AF are internal bisectors of ∠BAC and ∠DAC respectively

Required: To prove EF||BD

Consider the ΔABC,

Here, AE is the angle bisector of ∠A

∴ By angle bisector theorem --eq(1)

Now, In ΔACD,

Here, AF is the angle bisector of ∠A

∴ By angle bisector theorem

⇒ (∵ AD = AB) --eq(2)

From –eq(1) and –eq(2)

We have,

Now, Consider ΔBCD

Here,

∴We can say that EF||BD by Converse Thales theorem.

Hence Proved