We report on controllable cavity modes by controlling the backscattering by two identical scatterers. Periodic changes of the backscattering coupling between two degenerate cavity modes are observed with the changing angle between two scatterers and elucidated by a theoretical model using two-mode approximation and numerical simulations. The periodically appearing single-peak cavity modes indicate mode degeneracy at diabolical points. Interactions between single quantum dots and cavity modes are then investigated. Enhanced emission of a quantum dot with a six-fold intensity increase is obtained in a microdisk at a diabolical point. This method to control cavity modes allows large-scale integration, high reproducibility and flexible design of the size, the location, the quantity and the shape for scatterers, which can be applied for integrated photonic structures with scatterer-modified light-matter interaction.