  Regarding the geometry of a real normed space ${\mathcal X}$, we mainly
introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in
{\mathcal X}$ as follows: $${x\np{\varepsilon}}_W y \mbox{if and only if}
\sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq
\Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$
We study class of linear mappings preserving the approximately
bisectrix-orthogonality ${\np{\varepsilon}}_W$. In particular, we show that if
$T: {\mathcal X}\to {\mathcal Y}$ is an approximate linear similarity, then
$${x\np{\delta}}_W y\Longrightarrow {Tx \np{\theta}}_W Ty \qquad (x, y\in
{\mathcal X})$$ for any $\delta\in[0, 1)$ and certain $\theta\geq 0$.
