The dynamic configuration below is a special case of the Van Aubel Twin Circles configuration for similar parallelograms on the sides. Below the two twin circles coincide as well as some of the pairs of points of the general case.
If only X is dragged to change the shape of the similar rhombi, note how the two coincidental circles remain stationary as well E, and F, but the other points move on the circle. Also drag X to the other side of AB to observe the movement of points.
Van Aubel Coinciding Circles for Similar Rhombi
Created by Michael de Villiers, 10 November 2020 with WebSketchpad.