A less symmetric example starts with the xy plane in $\mathbf R^3,$ then introducing a radially symmetric hill with support within, say, the standard unit disk. The bisector of the points $(8,0)$ and $(10,0)$ is still the geodesic $x=9.$ However, the bisector of the points $(-2,0)$ and $(2,\frac{1}{2})$ is a little peculiar near the origin.