In parallelogram ABCD, the diagonal AC is the bisector of angle A. Find the side BC if the perimeter of ABCD is 128.

The CAD angle is equal to the ACB angle (internal criss-crossing angles with parallel BC and AD and secant AC).
The CAD angle is equal to the CAB angle (since AC is the bisector).
Consequently, the angle ACB is equal to the angle CAB. This means that triangle ABC is an isosceles triangle (in an isosceles triangle, the angles at the base are equal).
Hence, AB = BC. Since the opposite sides of a parallelogram are equal, it turns out that all sides of a parallelogram ABCD are equal. Hence, BC = P: 4 = 128: 4 = 32.
Answer: the BC side is 32.