Mathematics of college entrance examination: triangle AB=BC, AB = BC, where M is the midpoint of BC and AM is the bisector of angle BAC. Find the product of vector AB and vector BC.

Analysis: In the triangle ABC

∵ AB = BC, ∴⊿ABC is an isosceles triangle.

∵M is the midpoint of BC, and AM is the bisector of angle BAC.

According to the nature of the bisector of the inner angle, AC/AB=CM/MB= 1.

∴⊿ABC is an equilateral triangle

Let its side length be a.

∴ Vector AB Vector BC=| Vector AB|? Vector BC | cos60 = a 2/2