What is the difference in radii of two concentric circles given an angle and length of a triangle that is inscribed in the annulus?

Question

In relation to this geometric construction:

where D is the center of both circles, if the inner radius (x = length of line segments DA and DE), the angle φ = ∠CAB, and the length Δg of line segment AB are given, what is the length Δx of line segment AC?

Answer 1

Consider the triangle DAB. You know two sides and an angle. The length of the third side is $x + \Delta x$.