The aim of this talk is twofold: rstly, to establish a Fan type geometric result and to apply it in order to obtain some coincidence-like theorems for the case when the images of the correspondences are not convex. Further, new theorems concerning the existence of solutions for equilibrium problems are provided. For coincidence theorems, the reader is referred, for instance, to [2], [3], [10], [11], [13], [19]. The equilibrium problems have been studied, for example, in [1], [2], [3], [6], [7], [9], [12], [18]. Our goal is also to investigate whether the class of minimax inequalities can be extended. In fact, we obtain a new general minimax inequality of the following type: infx2Xsupy2Y t(x; y)  supy2Y infz2Z q(y;z) infz2Z supx2X p(x;z) : Its study is motivated and inspired by the results obtained, for instance, in [1], [2], [3], [19], which concern the three-function inequality: inf x2Xt(x; x)  supy2Y infz2Z q(y; z) + supz2Z infy2Y p(z; y). We intend to connect, in forthcoming papers, the present results with the new ones, which consider the equilibrium in games, and are established in [15] or [16]. Another recent result, a contribution of the author, regarding minimax inequalities for discontinuous correspondences, is [14]. In this rst part, the originality consists of introducing a new type of properly quasi-convex-like correspondences, which proved to play an important role in our results. The method of proof is based on the well known KKM property.