In triangle ABC, AB = BC, BD = DC, AC = 2 cm and AD is the bisector of angle A. Find BD.

Let’s deal with this task.
Given:
ABC – triangle
AB = BC
BD = DC
AC = 2cm
To find:
BD =?

Decision:
If AB = BC, then the triangle is isosceles.
The angles at the base are equal (by condition).
AD is a bisector that divides ∠A in half and at the same time BD = DC, so AD is also the median.
The bisector of the angle at the base of an isosceles triangle divides the lateral side into segments proportional to the lateral side and the base.
AB / AC = BD / DC
BD = DC, it means AB / AC = 1
AB = 1 * AC = 2 cm.
AB = AC + BC = 2 cm
Hence, ΔABS is equilateral.