In the given figure, AD is the internal bisector of ∠A and CE || DA. If CE meets BA produced at E, prove that △CAE is isosceles.

Given,

CE || AD

BE is the transversal.

From figure,

⇒ ∠DAC = ∠ACE …(1) (Alternate angles are equal)

⇒ ∠BAD = ∠CEA (Corresponding angles are equal)

But, ∠BAD = ∠DAC (as AD is bisector of ∠BAC)

⇒ ∠DAC = ∠CEA ….(2)

From eq.(1) and (2), we have:

∴ ∠ACE = ∠CEA

AE = AC (Sides opposite to equal angles in a triangle are equal)

∴ △CAE is isosceles triangle.

Hence, proved that △CAE is isosceles.