Question

In triangle ABC AD is a bisector of angle A and D is a mid-point of BC. Prove that is an isosceles triangle.

Answer 1

In triangle ABD and triangle ABC 
BD = DC ( AD is the bisector of side BC)
AD = AD (AD is the common side)
angle BAD = angle CAD (AD is the angle bisector of angle A)
Therefore triangle ABD is congruent to triangle ADC
which implies AB = AC
and angle B = angle C (angles opposite to equal sides are equal)
since 2 sides are equal and angles opposite to these equal sides are equal we can say triangle ABC in an isosceles triangle (by SAS congruency)