CC-version of Heron's formula (Q1026906)

Let \(\mathbb{R}^2_C= (\mathbb{R}^2, d_C)\) be the Chinese Checkers plane (CC-plane), where distance \[ d_C(P_1, P_2):= \max\{|x_1- x_2|, |y_1- y_2|\}+ (\sqrt{2}- 1)\min\{|x)1- x_2|, |y)1- y_2|\} \] for points \(P_1= (x_1, y_1)\), \(P_2= (x_2, y_2)\). This is a very special Minkowski plane. The group of isometrics is described by \textit{R. Kaya}, \textit{Ö. Gelişgen}, \textit{S. Ekmekçi} and \textit{A. Bayar} [Missouri J. Math. Sci. 18, No. 3, 221--233 (2006); Zbl 1184.51021)]. It is of interest to investigate well-known theorems of the elementary Euclidean geometry with regard to the CC-plane geometry. The presented CC version of the well-known Heron's formula consists of a lot of formulas with long numerous cases and subcases. This result is a little disappointing.