\frac{AI }{IB}=\frac{AB}{AI} 

A = (-1.2909090909090892, 1.6423140495867767)
B = (5.866115702479339, 1.6588429752066114)
a = Segment(A, B)
k = 1.618033988749895
c: a / k
d = Circle(A, c)
I = Intersect(d, a, 1.0)