Question

If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

Answer 1

Produce AD up to E such that AD = DE.
In ΔABC and ΔEDC,
AD =DE ........[by construction]
BD = CD ...........[Given]
∠1 = ∠2 ..........[verticaly opposite angles]

∴ ΔABD ≅ ΔEDC ......[SAS]
⇒ AB = CE ........(i)
and ∠BAD = ∠CED
But, ∠BAD = ∠CAD .......[AD is bisector of ∠BAC]
∴ ∠CED = ∠CAD
⇒ AC = CE ........(ii)
From (i) and (ii)
AB = AC
Hence, ABC is an isosceles triangle.