A New Generalization of the Steiner Formula and the Holditch Theorem

Abstract

In this study, we first obtained the Steiner area formula in the generalized complex plane. Then, with the aid of this formula, we determined a new approach for the Holditch theorem giving the relationship between the areas formed by points in the generalized complex plane (or p-complex plane). Finally, according to the special values of p = −1, 0, 1 we examined the cases of the Steiner Formula and Holditch Theorem. In this way, for \({p \in \mathbb{R}}\) we generalized the Steiner Formula and Holditch theorem consisting the Euclidean \({\left({p = -1}\right)}\), Galilean \({\left({p = 0}\right)}\) and Lorentzian \({\left({p = 1}\right)}\) cases.