This paper presents a comprehensive study of two geometric constants, \(\theta _I(X)\) and \(\theta _B(X)\) , defined on a Banach space X using isosceles and Birkhoff orthogonality, respectively. These constants are defined by \( \theta _I(X) = \sup \left\{ \frac{\Vert \Vert x+y\Vert x - (x+y) \Vert }{\Vert x+y\Vert } : x, y \in S_X, \, x \perp _I y \right\} , \) \( \theta _B(X) = \sup \left\{ \frac{\Vert \Vert x+y\Vert x - (x+y) \Vert }{\Vert x+y\Vert } : x, y \in S_X, \, x \perp _B y \right\} , \) where \(S_X\) is the unit sphere, and \(\perp _I\) and \(\perp _B\) denote isosceles and Birkhoff orthogonality. We establish new inequalities relating these constants to well-known geometric parameters such as the James constant J(X), the Schäffer constant S(X), and the von Neumann-Jordan constant \(C_{NJ}(X)\) . Furthermore, we derive sufficient conditions in terms of \(\theta _I(X)\) for a Banach space to possess the fixed point property. Our results can be viewed as a further extension and exploration of the orthogonal geometric constants defined by Xie et al. [11], offering deeper insights into the geometric properties of Banach spaces via these newly introduced constants.
