A characterisation of Euclidean normed planes via bisectors

Our main result states that whenever we have a non-Euclidean norm $\|\cdot\|$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $\lambda\neq 1, \lambda>0$, there exist $y, z\in X$ verifying that $\|y\|=\lambda\|x\|$, $z\neq 0$, 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.