The aim of this article is to study the new geometric constant \(C_{B}(X)\) , which is on account of the ideas of the parameter D(X) introduced by Ji and Wu (J. Math. Anal. Appl. 323, 1–7, 2006). First, it is shown that the constant provides us a new way to check the discrepancy between Birkhoff orthogonality and isosceles orthogonality as well as measures the cosine between two vectors in the right triangles. Next, we make use of some inequalities to establish the relationships between some existing geometric constants and \(C_{B}(X)\) . Meanwhile, we also discuss some applications of the constant \(C_{B}(X)\) . To study the constants related to the sine and the cosine between two vectors in the equilateral triangles, we also treat \(S_{\Delta }(X)\) and \(C_{\Delta }(X)\) . In particular, we give a characterization of uniformly non-square Banach spaces in terms of \(S_{\Delta }(X)\) . In the end, we also study the relationship between \(S_{\Delta }(X)\) and the fixed point property for nonexpansive mappings.
