The nonlocal geometric variational problem derived from the Ohta-Kawasaki diblock copolymer theory is an inhibitory system with self-organizing properties. The system can prevent a disc from drifting towards the domain boundary. This raises the question whether a stationary set may have its interface touch the domain boundary. It is proved that a small, perturbed half disc exists as a stable stationary set, where the circular part of its boundary is inside the domain, as the interface, and the almost flat part of its boundary coincides with part of the domain boundary. The location of the half disc depends on two quantities: the curvature of the domain boundary, and a remnant of the Green's function after one removes the fundamental solution and a reflection of the fundamental solution. The notion of reflection here is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion. Our analysis of a boundary half disc leads to constructions of stationary assemblies with both interior discs and boundary half discs.