I imagined rotating the figure such that AC was the vertical axis and its bisector was the horizontal axis, and tried to imagine some way to perturb D and E to get a sequence of quadrilaterals that wouldn't be congruent (because the angles ∠CDA and ∠CEA were changing), but for which we could alternately take ABCD and ABCE so that successive shapes in the sequence would be partners. I couldn't see a way to make it work. Then I thought, what if perturb B instead?
I imagined rotating the figure such that AC was the vertical axis and its bisector was the horizontal axis, and tried to imagine some way to perturb D and E to get a sequence of quadrilaterals that wouldn't be congruent (because the angles ∠CDA and ∠CEA were changing), but for which we could alternately take ABCD and ABCE so that successive shapes in the sequence would be partners. I couldn't see a way to make it work. Then I thought, what if perturb B instead?