Poncelet closure theorem, which is a theorem about closing property for a sequence of polygons inscribed in one and circumscribed about the other given conic, is one of the most explored and most important theorems of projective geometry. Besides Poncelet theorem, there are more theorems of this type, such as Steiner theorem, "Zigzag" theorem and Emch theorem, which is in a sense the strongest one among these four. At the beginning, we present a generalized closing property on a circle \(k\) in the plane and later on a corresponding property is formulated for spheres and a given circle \(k\) in \(d\)-dimensional Euclidean space. It is proved that any family of spheres \(\mathcal{F}\) , that satisfies certain nonsingularity conditions and a simple analytical relation, possesses the closing property on a circle. In that case, the centres of all spheres from \(\mathcal{F}\) belong to a certain quadric \(\Gamma\). The geometrical meaning of this condition is that spheres from \(\mathcal{F}\) are tangential to two spheres \(S_0\) and \(S_1\) and orthogonal to a given sphere \(Q\). It is shown that four classical theorems of Poncelet type are corollaries of the general closing property for spheres. In the last section, we focus on the elementary proof of Emch’s theorem in the plane and its extension to the Euclidean space \(\mathbb{R}^d\).