Is there a Riemannian metric $g$ on $\mathbb{R}^2$ with inducing distance $d$ which is not isometric to the standard metric but satisfy the property quoted bellow?

For every two distinct points $p,q\in \mathbb{R}^2$, the locus of all points $z$ with $d(z,p)=d(z,q)$ is a geodesic.

As a higher dimensional version:

Is there a Riemannian metric on $\mathbb{R}^3$ which is not isometric to the standard metric but satisfy the property quoted bellow?

For every two distinct points $p,q\in \mathbb{R}^3$, the locus of all points $z$ with $d(z,p)=d(z,q)$ is a 2 dimensional submanifold with constant sectional curvature?

# Edit:

According to the comment of Willie Wong, I realized that the previous version of the question should be reconstructed. So I present the question as follows:

Assume that we have a complete Riemannian metric on $\mathbb{R}^n$ which satisfies the following property: For every $2$ points $p,q$, the locus of all points $\{z\mid d(z,p)=d(z,q)\}$ is a codimension- $1$ smooth submanifold which is a totally geodesic submanifold. Does this imply that the metric is isometric to either Hyperbolic space or the Euclidean space?