BD is the bisector of angle ABC. AB = BC prove that triangle ABD = CBD.

Since triangle ABC, according to the condition, is isosceles, the bisector of angle ABC is also the height of the triangle and its median, then AD = CD = AC / 2, and triangles ABD and BCD are rectangular.

In an isosceles triangle, the angles at the base are equal, angle BAC = BCA, side AB = CA.

Then right-angled triangles ABD and CBD are equal in hypotenuse and acute angle, the third sign of equality of right-angled triangles, which was required to prove.