Angle Bisector

The circles at D and E 
must have the same radius.

A = (2.94, 4.48)
B = (-1.04, 1.34)
C = (4.18, -0.72)
a = Segment(A, B)
b = Segment(B, C)
c = Circle(B, 0.5 Distance[A, B])
D = Intersect(c, a)
E = Intersect(c, b)
d = Circle(D, 0.65 Distance[B, D])
e = Circle(E, 0.65 Distance[B, D])
F, G = Intersect(d, e)
f = Ray(B, F)
α = Angle(F, B, A)
β = Angle(C, B, F)