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

Given: ABC is a triangle and D is an angle bisector such that BD = DC.

To prove: ABC is an isosceles triangle.

Proof: BD = DC and ∠BAD = ∠CAD

Consider Δ ABD and Δ ADC,

AD = AD (Common)

∠BAD = ∠CAD (Given)

BD = DC (Given)

By SAS congruency criterion,

Δ ABD ≅ Δ ADC

By corresponding parts of congruent triangles,

AB = AC

Hence, Δ ABC is isosceles triangle.

Hence, if the bisector of an angle of a triangle bisects the opposite side, the triangle is isosceles.