A characterisation of Euclidean normed planes via bisectors

Our main result states that whenever we have a non-Euclidean norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ on a two-dimensional vector space $X$, there exists some $x\mathrm{\ne }0$ such that for every $\lambda \mathrm{\ne }1,\lambda >0$, there exist $y,z\in X$ verifying that $\mathrm{\parallel }y\mathrm{\parallel }=\lambda \mathrm{\parallel }x\mathrm{\parallel }$, $z\mathrm{\ne }0$, and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$. Throughout this paper we also state and prove some other simple but maybe useful results about the geometry of the unit sphere of strictly convex planes.