Abstract: The cone theorem is one of the fundamental theorems of the minimal model program (MMP). The theorem states that the part of Mori’s cone whose intersection numbers are negative with the canonical divisor is locally generated by finite extremal rays.In the dual space, the cone theorem can be described in terms of divisors, which means the visible boundary of the nef cone with respect to the canonical divisor is a rational polytope, and those extremal rays correspond to the faces of the rational polytope. The dual cone theorem can be proved by assuming the finite generation of the canonical rings. By observing the above duality, a more geometric proof of the dual cone theorem is given without assuming the finite generation of the canonical rings.