Before revision:  The proofs of non-circularizability use various techniques, most depend on incidence theorems , others use arguments involving metric properties of arrangements of planes, or angles in planar figures .
After revision:  The proofs of non-circularizability use various techniques, most depend on incidence theorems like Miquel's. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way .