Question

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

Answer 1

Given: In triangle ABC, AD is the bisector of angle A and AD also bisects side BC, so D lies on BC with BD = DC.

To Prove: AB = AC (i.e., triangle ABC is isosceles).

Proof (Step-wise):

1. AD bisects angle A, so ∠BAD = ∠CAD.   ...(Given)

2. AD bisects BC, so BD = DC.   ...(Given)

3. By the Angle Bisector Theorem, an internal angle bisector divides the opposite side in the ratio of the adjacent sides.

Hence `(AB)/(AC) = (BD)/(DC)`.

4. From step 2,

BD = DC.

So, `(BD)/(DC) = 1`.

Therefore, `(AB)/(AC) = 1`, which implies AB = AC.

AB = AC, so triangle ABC is isosceles.