AB = BC, angle A is 60 degrees, CD is the bisector of angle BCE. Prove that AB // CD.

Since, by condition, AB = BC, then the triangle ABC is isosceles, then the angle BCA = BAC = 60.

Then the angle ABC = (180 – BAC – BCA) = (180 – 60 – 60) = 60, and therefore the triangle ABC is equilateral, AB = BC = AC.

Consider a triangle ACD in which the angle CAD = 60, and the angle ACD = ACB / 2 = 60/2 = 30, since CD is the bisector of the ACB angle.

Then the angle АDС = (180 – АСD – CAD) = (180 – 30 – 60) = 90.

Consequently, AB is perpendicular to CD, as required.