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Geometry projection-based topology optimization has attracted a great deal of attention because it enables the design of structures consisting of a combination of geometric primitives and simplifies the integration with computer-aided design (CAD) systems. While the approach has undergone substantial development under the assumption of linear theory, it remains to be developed for non-linear hyperelastic problems. In this study, a geometrically non-linear explicit topology optimization approach is proposed in the framework of the geometry projection method. The energy transition strategy is adopted to mitigate excessive distortion in low-stiffness regions that might cause the equilibrium iterations to diverge. A neo-Hookean hyperelastic strain energy potential is used to model the material behavior. Design sensitivities of the functions passed to the gradient-based optimizer are detailed and verified. The proposed method is used to solve benchmark problems for which the output displacement in a compliant mechanism is maximized and the structural compliance is minimized.